Why is The Tempest seen as Shakespeare's theater farewell?

Whether or not The Tempest can be construed as Shakespeare's farewell to the stage is a matter of considerable dispute. However, both in chronological and thematic terms, the play could indeed be interpreted in such a light. For one thing, The Tempest is generally accepted as being the last full play that Shakespeare ever wrote by himself; he subsequently collaborated on two further plays before his death. Thematically speaking, it is noteworthy that the play consists of a magician giving up the practice of his art. Much scholarly attention has been focused on the highly suggestive words spoken by Prospero in act 4, scene 1:

And like the baseless fabric of this vision, The cloud-capped towers, the gorgeous palaces, The solemn temples, the great globe itself—Yea, all which it inherit—shall dissolve, And like this insubstantial pageant faded, Leave not a rack behind. We are such stuff As dreams are made on, and our little life Is rounded with a sleep. [Emphasis added].

Some have suggested that Shakespeare could be referring here to the Globe Theatre, built by Shakespeare's playing company, the Lord Chamberlain's Men. Just as Prospero is turning his back on the things of this world, so the argument runs, so too is Shakespeare preparing himself to say farewell to the profession in which he has excelled for so long.

Do you agree that The Tempest is Shakespeare's farewell play? Why?

Literary critics and Shakespeare experts almost uniformly hold the opinion that The Tempest is Shakespeare's "farewell play." This opinion is based upon three major points. The first major reason is that this was the last play that Shakespeare wrote completely himself, though he is said to have collaborated with John Fletcher on The Two Noble Kinsman (an adaptation of Chaucer's "The Knight's Tale") c. 1621. 

The second major reason is the thematic similarities between Prospero as a magician and Shakespeare as a playwright. Briefly, Prospero stage directs the tempest that moves the plot of his own personally contrived drama and then commands the entrances and exists of each of the individuals whose lives he is manipulating. He does this with the help of his own personal stage manager, Ariel. Thus Prospero's abilities and actions in this encounter between the opposing factions of the play are akin to and dramatizations of Shakespeare's role as a playwright who does precisely the same things but on paper.

The third is posited by Shakespearean director Trevor Nunn. He points to evidence in the play's original program that indicates fifty-year-old Shakespeare played fifty-year-old Propsero. "William Shakespeare" is at the top of the list of players and "Prospero" is at the top of the list of Characters. Shakespeare may have in Nunn's words, said farewell to acting and to writing at the same time by acting the role of Prospero himself.

What is significant about The Tempest being Shakespeare's final play?

The substance of your question has been addressed in the first link below, "Did Shakespeare intend The Tempest to be his last play?"

There are, however, a few more hints of what might be called a sense of finality in The Tempest than the above posting references. Prospero's final speech seems to be a farewell to more than the play -- it is dramatically unnecessary for him to beg the audience for mercy, and say that his mission was only to entertain them.

Gentle breath of yours my sails
Must fill, or else my project fails,
Which was to please. Now I want
Spirits to enforce, art to enchant;
And my ending is despair,
Unless I be reliev'd by prayer,
....
As you from crimes would pardon'd be,
Let your indulgence set me free. (Epilogue, my emphasis)

Moreover, this tone is shared by some of the things he says earlier, including the most famous lines in the whole play:

Our revels now are ended. These our actors,
As I foretold you, were all spirits and
Are melted into air, into thin air:
And, like the baseless fabric of this vision,
The cloud-capp'd towers, the gorgeous palaces,
The solemn temples, the great globe itself,
Yea, all which it inherit, shall dissolve
And, like this insubstantial pageant faded,
Leave not a rack behind. We are such stuff
As dreams are made on, and our little life
Is rounded with a sleep. (IV, i)

Thus, gently contra the argument referenced below, I think that The Tempest is indeed meant as a farewell by Shakespeare to his career in the theatre.