Dimmesdale dies on the scaffold shortly after he and Hester make plans to run away together and escape Boston and the judgment of their society. He calls Hester and Pearl up on to the scaffold with him as if to publicly acknowledge his role in Pearl’s creation, and he must physically lean on Hester’s shoulder because he is so weak.

He exclaims to all present that he ought to have stood upon the scaffold with Hester seven years earlier, when she received her punishment for adultery, and he says that his “brand of sin and infamy” has been hidden from everyone in the town all this time. He struggles with a “bodily weakness” and a “faintness of heart” that seems to nearly overcome him.

Dimmesdale declares that his “own red stigma” has, in fact, “seared his inmost heart!” He then tears away his shirt and ministerial collar to reveal something shocking, though that something is never explicitly named. It is likely his own scarlet “A” that he has somehow either burned or carved into his own skin—the something that Chillingworth saw years before that convinced him that Dimmesdale was Hester’s lover. Just then, Dimmesdale sinks to the ground and requests Pearl’s kiss, which she gives to him. He says goodbye to Hester, finally, and dies.

You'll remember that in The Scarlet Letter, when Hester commits adultery, she refuses to name the father of her unborn child.  It is Dimmesdale that argues, after Pearl's (the baby's) birth, that Hester should be allowed to remain in the community as her mother.

When Hester's long-lost husband, Chillingworth, appears in disguise, he befriends Dimmesdale, though they are very different, and with his knowledge and background, passes himself off as a physician. (His intent is revenge on the man that Hester was unfaithful with.)  Dimmesdale is unwell, often putting his hand to his heart; his guilt eats away at him over the years, for his is the father of Hester's child.

At one point (while Dimmesdale is in a deep--perhaps drugged--sleep), Chillingworth examines the minister; seeing something beneath the sick man's shirt, Chillingworth's need for revenge and his hatred intensify.

By this time, Pearl is seven.  Hester talks to Chillingworth and asks him to stop tormenting Dimmesdale, realizing why her "husband" is staying so close to the minister.  Chillingworth refuses, so Hester tells Chillingworth's secret to Dimmesdale, and the couple agree to leave, taking Pearl with them, to start a new life.  Chillingworth discovers their plan.

The pair now know they will never be free of Chillingworth.  As he leaves the church after services, suddenly, Dimmesdale's vitality leaves him, and clutching at Hester and Pearl, he confesses his part in Hester's fall from grace, shows the mark hidden beneath his shirt, which looks like an "A," and Dimmesdale dies.

The "A" he "wears" ties him to the adulterous affair with Hester, though she had never disclosed his identity in all the years in which she lived as an outcast in this Puritan environment.

In Nathaniel Hawthorne's The Scarlet Letter, Arthur Dimmesdale suffers the effects of guilt from the very first pages.  When we first meet him, he is speaking to Hester--only as her pastor, of course--and even then he has the habit of clutching his hand to his heart as if in pain.  As the story progresses, he gets weaker and more frail, still continually clutching his hand to his heart.  He is pale and he sweats, and we just know he's not well.  Roger Chillingworth understands that whatever is deteriorating his body is intricately connected to whatever is eating away at his soul.  He discovers something on Dimmesdale's chest (presumably some version of Hester's scarlet letter) while he is sleeping soundly, and we understand he is the adulterer complicit with Hester Prynne. We learn he's been keeping vigils (not sleeping), fasting (not eating), and plying a bloody scourge (whipping himself).  

Arthur has a renewed sense of vigor and purpose after he and Hester talk in the forest and make plans to leave.  On Election Day, Dimmesdale walks through town in the best health we've seen since we first meet him in chapter 2.  After he has delivered the sermon, though, he leaves the church an even more broken and frail man.  After he climbs the scaffold with Hester's help, he makes peace with Pearl and with God and confesses publicly.  After doing so, he simply dies.  All of that to get to your question: How did he die?  That's what happened; why it happened is probably because he had simply worn his body out.  Because his external self was so connected to his internal self, when he finally let go of his guilt, he was free to let go of his weakened physical self, as well.  There is really no other satisfying explanation for his sudden death. 

Reverend Dimmesdale dies of an illness that haunts his soul.  Tormented by his terrible guilt and the psychological torture of Roger Chillingworth who has vowed that "he will be mine," Dimmesdale is so burdened spiritually that his health fails and he can bear his sin no more.  In Chapter XIX of The Scarlet Letter, Hawthorne writes,

So it ever is, whether thus typified or no, that an evil deed invests itself with the character of doom.

In his hypocrisy born from his guilt, Dimmesdale has been forced to wear a masque before his pashioners, and, as Hawthorne observes,

No man, for any considerable period can wear one face to himself, and naother to the multitude, without finally getting bewildered as to which may be the true.

After the failed attempt of Hester to procure passage for them to escape to England, Dimmesdale despairs of freeing himself from his guilt in any other way other than fully revealing his sin by climbing onto the scaffold and confessing and displaying his own letter of adultery.  His body so weakened by his guilt collapses, and he dies, having at least freed his soul by taking the shame of Pearl and Hester upon himself--"How feeble and pale he looked, amid all his triumph!"