There are some phrases — like “I hate to say it”, “with all due respect”, and “it’s not about the money, but…” — that sound honest but signify that the speaker is actually lying.
The point of a but-head is to preemptively deny a charge that has yet to be made, with a kind of “best offense is a good defense” strategy. This technique has a distinguished relative in classical rhetoric: the device of procatalepsis, in which the speaker brings up and immediately refutes the anticipated objections of his or her hearer. When someone says “I’m not trying to hurt your feelings, but…” they are maneuvering to keep you from saying “I don’t believe you — you’re just trying to hurt my feelings.”
See also the non-apology apology.