the function of "would" in a particular sentence

Historically, would is the past tense of will, and will had a meaning of "be willing to" (in fact it underlies the word "willing").

This meaning is rare in Modern English, but this idiom "would only say" is a case where it survives: it means "was willing to say ... and no more".

It means either

a) that whenever the conversation turned to politics, H. said only that one must get involved, or

b) that when the conversation turned to politics, H. said that one must get involved, but when prodded to be more forthcoming than that, and asked to give a more specific opinion, he was not more forthcoming.

b) is the more likely here, given Lilla's "When the conversation turned, inevitably..." for if he meant that the conversation always seemed to turn to politics, he would probably have used the word "returned" instead of "turned".