What truths about himself does Hamlet discover at the end of the play?

For me, this is one of the most interesting things about
the play, and brings up two of the most poignant quotes of the play.  Hamlet has spent
the entire play "thinking" about his actions and his lack of action.  He has berated
himself for not acting.  He has talked to the players about how to act, to his mother
about how to act, and even to Ophelia.  He has tried to assure himself that the ghost
was telling the truth and that Claudius did, indeed, kill King Hamlet.  Through all of
his, Hamlet is trying to keep control of himself and his circumstances.  What he
realizes by the end of the play is that isn't possible to have absolute control.  In a
very important conversation with Horatio, Hamlet tells him about what happened on the
boat to England.  Before he gets into the details of the story, he says that there are
times

When
our deep plots do pall; and that should learn us

There's a
divinity that shapes our ends,

Rough-hew them how we
will.

What Hamlet has come to
realize is that no matter what choices we make to shape our lives, there is all kinds of
fate that also play a part in our lives, and there is absolutely nothing we can do to
control that.  All we can do, he says later in Act 5, is be ready to act when the time
(fate) comes.  He is specifically talking about the upcoming fencing match with
Laertes.  Horatio is concerned that this will be dangerous, but Hamlet has stopped
thinking he can control everything.  Fate has brought him to this place -- the showdown
with Laertes, with the King and Queen present.  All he can do is be as ready as possible
for what may come of this.  I don't think we are to read the lines as fatalistic, but
just realistic.  Hamlet is not looking to die; Hamlet doesn't assume the worst; but he
is ready for whatever comes his way.  When tells Horatio his thoughts he even alludes to
the gospel of Matthew saying:

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We defy augury.  There is special providence
[fate,desitiny]in the fall of a sparrow.  If it [death, vengeance] be now, 'tis not to
come; if it be not to come it will be now; if it be not now, yet it will come.  The
readiness is all.

These ideas
are what allow Hamlet to stop thinking and planning and checking himself,and to just act
-- to just go into the fencing match with the best intention of winning and being ready
for whatever may come his way.