these words entail is that her actions, and the consequences of them, have changed Hester so tremendously that she basically was reborn again into another woman. Although this woman has less than an ideal life, it is life nevertheless and Hester felt a moral responsibility to embrace it and live with it. Therefore, that first reason can be attributed to moral responsibility to abide by whatever comes her way. 

The second reason is Dimmesdale. 

another feeling kept her within the scene [...] one with whom she deemed herself connected in a union, that, unrecognized on earth, would bring them together before the bar of final judgment...

Even after the humiliation, the loneliness, and the terrible treatment that Hester has endured from the villagers and the aldermen alike, the fact remains that Hester not only feels love for Dimmesdale, but also a spiritual connection that keeps her bound to him. It is almost unbelievable to the modern reader that Hester would harbor any love inside of her for the man that has left her to suffer in silence for actions committed by the two of them. Moreover, that this man is faking his righteousness and still acts like a man of God makes her connection to him all the more awkward. However, those are exactly the reasons why Hester decides to continue a sad life in the village facing society head on.

It is curious, isn't it, that after her daughter inherits so much money and is able to leave for the Old World, that her mother decides to stay in the colony where her life and name are marred by the scarlet letter. And yet, as Hawthorne tells us in the final chapter of this amazing novel, in a sense, it is perfectly logical for her to stay in the colony:

But there was a more real life for Hester Prynne here in New England than in that unknown region where Pearl had found a home. Here had been her sin; here her sorrow; and here was yet to be her penitence. She had returned, therefore, and resumed - of her own free will, for not the sternest magistrate of that iron period would have imposed it - resumed the symbol of which we have related so dark a tale.

Thus we are told that Hester Prynne remained in the colony, in spite of so many reasons to leave, because she felt it was in the colony that she should still serve penitence for her "sin." Her entire experience of sin, sorrow and shame had been based in the colony, and she presumably felt that after these intense experiences she would be unable to start again somewhere else.

For one, she has nowhere else to go.  Two, she believes that running would only convince the people further of her guilt, which she does not believe. 

Later in the novel, her "A" comes to mean more than "adultery"; she proves herself an "able" and productive member of the colony;  an "asset" rather than an embarrassment.

Twice one wonders why Hester lives in a village so eager to banish her from their companionship.

The first occurs shortly after her punishment on the scaffold and the second when she returns to end her days in her former cottage.

The first time Hawthorne addresses this is in Chapter 5:

It may seem marvellous, . . . that this woman should still call that place her home, where, and where only, she must needs be the type of shame. But there is a fatality, a feeling so irresistible and inevitable that it has the force of doom, which almost invariably compels human beings to linger around and haunt, ghost-like, the spot where some great and marked event has given the color to their lifetime; . . . It might be, too,. . . that another feeling kept her within the scene and pathway that had been so fatal. There dwelt, there trode the feet of one with whom she deemed herself connected in a union, that, unrecognized on earth, would bring them together before the bar of final judgment, and make that their marriage-altar

This passage is heavily excerpted but indicates a two-fold reason for young Hester to remain. She feels bound to this site since it was the place of her momentous sin, and she harbors a secret hope that she and Arthur may be united in some afterlife. This passage hearkens back to Hawthorne's own musing in The Custom House when he discusses his own return to Salem, the sight of his family's ancestral guilt:

It is no matter that the place is joyless for him; that he is weary of the old wooden houses, the mud and dust, the dead level of site and sentiment, the chill east wind, and the chillest of social atmospheres;—all these, and whatever faults besides he may see or imagine, are nothing to the purpose. The spell survives, and just as powerfully as if the natal spot were an earthly paradise. So has it been in my case. I felt it almost as a destiny to make Salem my home; so that the mould of features and cast of character which had all along been familiar here,—ever, as one representative of the race lay down in his grave, another assuming, as it were, his sentry-march along the main street,—might still in my little day be seen and recognized in the old town.

Humans seem to have a rootedness in the soil that bears their shame, and Hester stays for this gloomy quality as well as her apparent lingering love for Arthur.

The second time we hear of Hester living here is at the end, when she returns from Europe to end her days in her little dwelling outside the town. Hawthorne says she spends her days in willfully meager existence, visited by others who seek some comfort for the suffering they experience in this world. In this capacity, she seems almost a prefigurement of the type of person Hawthorne prophesies will bring to America a greater hope and joy:

Hester comforted and counselled them as best she might. . . . Earlier in life, Hester had vainly imagined that she herself might be the destined prophetess, but had long since recognized the impossibility that any mission of divine and mysterious truth should be confided to a woman stained with sin. . . .The angel and apostle of the coming revelation must be a woman, indeed, but lofty, pure, and beautiful; and wise, moreover, not through dusky grief, but the ethereal medium of joy; and showing how sacred love should make us happy, by the truest test of a life successful to such an end!

The narrator tells us in the final chapter that "Here had been her sin; here, her sorrow; and here was yet to be her penitence."  In other words, Hester had chosen to return to Boston because this is the place where the most defining events of her life had taken place, and she feels the need to allow the remainder of her life to play out here. Yes, it had been the place of her sin, but it had also been the place of her love and her loss.  Here, in Boston, she can best remember that love and be close to it, even though Dimmesdale is dead and gone. The fact that, after Hester dies, she is buried next to Dimmesdale in the King's Chapel cemetery shows the connection she felt to him even after his death.

Further, the narrator says that "there was a more real life for Hester" in Boston than there could be anywhere else in the world.  Here, she has an identity, and people come to her -- especially women -- for advice and comfort because they know that she has been through so much difficulty and hardship. It seems not to feel honest to Hester for her to live her life somewhere else.  Her time in Boston has become such a part of her identity that she seems not to know how to live without that identity, and it allows her to actually do some good in the world.

In The Scarlet Letter, Hester Prynne chooses to stay in Boston out of attachment to the life-changing events that had happened there and to the person involved in them--the man who desecrated her marriage but was not punished with adultery charges.

The narrator describes this decision as "marvelous" in that there was absolutely no laws or stipulations of her punishment that forced her to remain there; it was completely a matter of free will. Hester has every opportunity to leave and is,

"free to return to her birthplace, or to any other European land, and there hide her character and identity under a new exterior, as completely as if emerging into another state of being."

Despite this, she does not run from her sin or from her identity; her sin, in fact, seems to give her "a new birth" which commits the land to her as a "life-long home." 

Meanwhile, she sees the man she cheated with as being connected to her in

"a union, that, unrecognized on earth, would bring them together before the bar of final judgement, and make that their marriage-alter."

The narrator describes this motive--Hester's desire to be weighed and judged by God with this "tempter of souls"--as "half a truth, and half a self-delusion." 

Overall, Hester decides that the place where she committed her greatest sin must be the place where she carries out her "earthly punishment." Therefore, she settles into "a small thatched cottage" and prepares to endure a great deal of ostracism from the community. 

Hester Prynne is one of the two primary characters in The Scarlet Letter by Nathaniel Hawthorne, and she is the wearer of the scarlet letter. After Arthur Dimmesdale dies, Roger Chillingworth has no reason to live, and he dies within the year. Soon after Chillingworth's death, Hester and Pearl actually do leave New England

[T]he wearer of the scarlet letter disappeared, and Pearl along with her. For many years, though a vague report would now and then find its way across the sea,--like a shapeless piece of driftwood tost ashore, with the initials of a name upon it,--yet no tidings of them unquestionably authentic were received. 

One day, without any warning or fanfare, however, Hester returns alone to her small house near the water. 

So perhaps a better question is why she left England and came back to this place which had known her shame. Of course she did not have anything substantive to draw her back here: no friends, no real connections of any kind. Yet she returns and resumes wearing the scarlet letter, despite the fact that no one would ever have required her to wear it. Pearl stayed in England (happily married and now "normal," from all accounts), and Hester could have stayed there, too. 

Hawthorne tells us why Hester came back to New England.

[T]here was a more real life for Hester Prynne, here, in New England, that in that unknown region where Pearl had found a home. Here had been her sin; here, her sorrow; and here was yet to be her penitence. She had returned, therefore, and resumed,--of her own free will, for not the sternest magistrate of that iron period would have imposed it,--resumed the symbol of which we have related so dark a tale. 

Hester came back here because she has a connection to this place because of Arthur and Roger and her sin. She obviously feels inextricably connected to this place for complicated reasons--love, hate, shame, guilt--and this, of course, is where she dies and is buried. 

In Nathaniel Hawthorne's The Scarlet Letter, Hester Prynne is forced to wear a scarlet "A" on her chest to display her shame to everyone with whom she interacts. She could easily slip away from her community, remove the "A", and live anonymously in a new community, either in the New World or in England. However, Hester chooses to stay, a decision that confuses many readers.

To understand her reasoning, the reader has to better understand Hester herself: she is not a rebel. Hester is a Puritan woman, and she believes in the doctrines of her fate. As a result, she believes that she has sinned and deserves to be punished. If she were to leave and live in anonymity, thereby escaping her penance, she would be continuing to sin. Instead, she chooses to carry out what she believes is a just punishment for her immoral crime. 

The narrator makes clear that Hester, her prison sentence over, is free to go anywhere she wants in the world: to Europe, to another colony, or even into the wilderness.

He attributes her remaining in Boston, the site of her shame, to "fate," stating that

. . . a feeling so irresistible and inevitable that it has the force of doom [. . .] almost invariably compels human beings to linger around and haunt, ghost-like, the spot where some great and marked event has given the colour to their lifetime; and, still the more irresistibly, the darker the tinge that saddens it.

Hester feels rooted to this place with what Hawthorne calls iron chains—not real iron chains, but a feeling in her soul that she must stay.

Part of this is that she hopes the "daily shame" of staying here, where everyone knows her crime, will purge and purify her soul, making her a better human being. She even dreams of becoming saintly through her suffering.

Hester does, however, move to the outskirts of town, to a small cottage, where she makes her living doing needlework. She lives very simply with Pearl and gives her extra money to help the poor.

I've thought about this on occasion, and I think it's because it's her home.  Although she had a strange life there, she had become "A"ble and an "A"ngel to many of the people.  She didn't have the same problems with what she did that Dimmesdale had; she had accepted her humanity and suggested that "What we did had a consecration of its own."

Her acceptance of herself, and her acceptancy by the community made it a comfortable place for her to be.

[I was unclear about whether this question pertain to her life before or after Dimmesdale's death.  If we are speaking of before, I'd have to add that her unquestioning love for Dimmesdale made it impossible to leave.  My other answer is about why she returns and stays there.]