The reason why the narrator cries is because he knows the truth: the reason why he taught Doodle to walk was not because he was being kind or helpful; instead, he teaches Doodle to walk because he says he is embarrassed by having a brother who couldn't walk. The narrator states, "They did not know that I did it for myself; that pride, whose slave I was, spoke to me louder than all their voices, and that Doodle walked only because I was ashamed of having a crippled brother."
This is an important development and observation by the narrator. There are several times in the story where this "streak of cruelty" shows itself, foreshadowing the end of the story. At one point, the narrator brings Doodle up in the top loft of the barn and shows him the casket that was meant for Doodle when everyone thought he would die - and then threatens to leave Doodle up there alone with the coffin. Doodle cries and pleads with the narrator not to leave him. This foreshadows the end of the story because Doodle screams for his brother not to leave him behind as Brother runs away in the rain, and Brother doesn't listen. Doodle ends up dying alone in the rain.
Unfortunately, it took the dying of Doodle, the "scarlet ibis" of a boy, for the narrator to recognize the error of his ways.