The modal auxiliary verb would was originally the past tense form of the modal will. As such, it has some of the same senses as will, specifically, in this case, that sense of "be willing to" that shows up when will is used in hypothetical clauses:
- if he will put down the gun,
which means if he is willing to put down the gun.
So the rather idiomatic question construction
means Why/Who/When/Where/.. would
X be willing to
Infinitive?, with the clear implication that the speaker is surprised at the possibility.
Conveying that surprise is the construction's most important function. Hence it's usually a rhetorical question.