James Fallows breaks it down. James Poniewozik also has some good analysis.
The conclusion? This solves nothing. And the fact that our President felt the need to subject himself to this means we all lose.
Update: I really appreciated the words of David Frum on this matter:
[T]hose who imagine that they somehow enhance the value of [American] citizenship by belittling the American-ness of their president – they not only disgrace the politics they uphold, but they do damage that will not soon be forgotten by the voters a revived Republicanism must win.
Salon also has some arguments on why this wasn’t Obama caving; it was Obama cannily portraying the Republican party as out of touch and a tiny bit crazy.