Interestingly, it seems that we changes meaning in the middle of the sentence.
Throughout the speech, we is used in a generic way, referring quite loosely to "we, the U.S. administration" and / or "we, the American people" or even "we, the world in general".
It is used in the same way I could ask "what do we call a long yellow fruit?"
However, the moment he mentions what Iraqis inside of Baghdad are thinking, the we he uses is part of the indirect speech (or rather, "reported thought"). We all of a sudden refers to the Iraqis.
It would have been more clear if he would have used direct speech (thought) like this:
But what we've also seen I think is a wake-up call for a lot of Iraqis inside of Baghdad recognizing: "we're going to have to rethink how we do business if we're going to hold our country together."
The only give-away for the change of meaning of we is the fact that "our country", which has to be kept "together", can, in this context only refer to Iraq, so people referring to it as our country have to be Iraqis.
Al in all, the sentence does not deserve a prize for clarity or style, in my opinion.