I would have to believe that the only logical way for my life to
go after the riots in 2001 was to dream-up a non-profit called
InkTank, to encourage anybody who ever wanted to set things on fire to
write down the deepest, most hidden part of their hearts. (Kathy
Is would have to the back-shifted form of will have to, or the non-past modal remoteness?
The sentence before the one you quoted is necessary to understand the usage:
Because if that were true, that there’s a plan, it would mean there’s a reason I live in Cincinnati, Ohio and my son robs banks. I would have to believe that …
Removing the intervening text, the important parts are “Because if that were true, … I would have to believe …”
I’m probably going to mess up the grammatical terms, but I think this is just the normal subjunctive mood, expressing the consequences of a contrafactual supposition.
Edit: On further reading, I think this usage is not subjunctive. Instead, it’s called a conditional, specifically the “second conditional”, according to the Wiki article.