What is very interesting about the sudden change of character that Edmund experiences is that the play suggests it was a result of the devotion of the two sisters Regan and Goneril for him. Note what he says after he hears the news that Goneril killed herself before poisoning Regan:
Yet Edmund was beloved:
The one the other poison'd for my sake,
And after slew herself.
Edmund, as is suggested by Kent's question, cries at the sight of the bodies of Regan and Goneril. After feeling an outcast and scorned and shunned throughout the entire play, he finally realises that he was sincerely loved, and he has the tangible proof in front of him. This seems to spur on the change of his character, as he says:
I pant for life: some good I mean to do,
Despite of mine own nature. Quickly send,
Be brief in it, to the castle; for my writ
Is on the life of Lear and on Cordelia:
Nay, send in time.
Realising that he himself is loved seems to make him reveal his essential goodness and to want to make amends for all that he has done in his life that has been reprehensible and evil. Thus one explanation would be to examine how the death of Regan and Goneril impact him and how their deaths act as a catalyst.