Evil can be atoned for to a certain extent by doing good, but the
evil cannot be "cancelled" without forgiveness. Forgiveness can only be
given by the one who was wronged, or by one who "absorbs" the wrong or
takes on the penalty in behalf of the shamed one.
Boromir is the honorable representative of his people. He succumbs to
unworthy behavior (the attempt to take the Ring) and allows evil to
take control. He is dishonored and disgraced because he is exposed as
one who has "defiled" himself, and has brought dishonor to his people,
and to his friend, Frodo. He is cut off from those with whom he
previously had fellowship, and he is cut off from his people because of
his dishonorable act. He is totally ashamed.
How can he atone for this? Up to a point, he can do something honorable
to offset the evil he has done; giving his life to save the kinsmen of
the one he dishonored does much to atone. But it is not enough to
cancel the guilt, he still needs to be forgiven by Frodo, or by one who
can act on Frodo's behalf.
Aragorn is the one who can grant this forgiveness. He is the true king,
and as king, he can take on the responsibility and the perogative to
forgive on Frodo's behalf. But he also has the authority to grant
forgiveness in the sense that he "absorbed" the evil and the dishonor
done to Frodo by Boromir, by himself successfully resisting the
temptation to take the Ring for himself and use it to get what he wants
most: his throne and all that comes with it. He also bears the fullness
of Boromir's shame by remaining silent about his confession, and he
does not tell the others about Boromir's dishonor (even in the film,
Gimli and Legolas do not arrive in time to hear the exchange between
the two as Boromir speaks of his dishonor as he dies), thus reducing
Boromir's dishonor in the eyes of others, even as he forgives him.
Frodo has been bearing the responsibility of the safety of the
Fellowship by keeping the Ring hidden, thus protecting the others from
its temptation. Because Aragorn has refused the Ring's call, Frodo
recognizes that he is worthy of complete trust. When he leaves to go to
Mordor alone, Frodo places the responsibility of caring for the others
into Aragorn's hands. In a way, Frodo has passed on his dishonor at the
hands of Boromir to Aragorn.
Thus, through Aragorn, Boromir can experience forgiveness as if Frodo
himself had given it, because Aragorn has now taken on this authority.
By his selfless act of atonement in fighting to save Merry and Pippin,
with Aragorn's pronouncement of forgiveness, Boromir's shame is now
removed and his honor is maintained. How satisfying it would have been
for Boromir, if he could have heard Frodo himself speak the words of
forgiveness! Nevertheless, he, too, recognizes Aragorn's authority to
grant forgiveness, and is content. He is able to die in peace, with a
smile on his lips.