The speaker starts this poem off by declaring that he's not going to give in to "Despair," a force that he labels "carrion comfort." He may be down and out, and the inner strands of his being may have gone slack, but he refuses to quit. He says that he can keep going. He's not going to end it all by killing himself (whew).
Still, he'd really like to know why a giant, crushing force like despair would be so rude as to smash him with its terrible power and subject him to violent storms. All he wants to do is make a break for it and get away from despair's awful influence.
The speaker doesn't wait around too long for an answer to his question, though. He's got a guess: despair is not crushing him; it's changing him—and for the better. By surviving his struggle with this awful force, the speaker's heart has grown stronger and more joyful.
Still, he's not sure where he should send the thank-you cards. Should he cheer for God for putting him through all this difficulty? Should he cheer for himself for surviving it all? Maybe both he and God are equally awesome in this scenario. Rather than coming to a clear decision, the speaker wraps things up by looking back at the "now done darkness" of his depression. He realizes that, not only was he struggling to overcome that crushing sadness, but he was also wrestling with God.