I don't know nearly enough yet about the details of the debt deal to say, but I have a sneaking suspicion that this is more or less how it went down in the end:
POTUS: "Look, you guys know that the Tea Party Republicans will block anything that they think I'm even a little tiny bit happy with. So let's throw together something fairly reasonable, and then all go out and pretend like I totally caved on everything so we can get the votes."
I mean, just on the surface it appears that the Tea Party has negotiated 10-year debt reduction from some $4 Trillion all the way up to $2.4 trillion -- well done! The two things that House conservatives absolutely demanded last week, actual passage of a balanced-budget amendment and major cuts to Pell grants, are both gone in the new deal. (On Pell grants, see here.) Entitlement reforms appear to be nixed, although possible for inclusion in "phase 2" later this year, if I understand correctly (and again, I'm admittedly working on minimal knowledge at this point), and the same is true of tax reform.
No question, this deal includes very significant cuts to domestic spending. And many conservatives will in fact vote against it. But I'm pretty sure Tea Party conservatives would be set on killing it if they weren't being told that this was a great victory over President Obama.