Well, the Church is infallible because God said so. Of course, the Church said that God said so. Of course, the Church only said that because God inspired them to say that.
What it comes down to is simply faith. Either you believe the Bible is the inspired Word of God, or you don’t. If you do, you must believe in the infallibility of the Church.
For me, I believe logic demands for there to be a God, and while maybe not necessarily the God we know of from the Bible, to me at least the God in the Bible is the most credible God I’ve ever heard of. I mean, seriously, why is there a universe? It makes absolutely no sense if there is no God for there to be anything at all. Accepting that, I’m forced to go looking for God. God probably cares about us, otherwise why bother making us? So I should look for a God that cares. A God that cares would have some kind of plan for us, and want us to know of Him, because again, why bother otherwise? Maybe it’s a God that we find through some kind of Cartesian thought exercise, but that’s not very comforting at all, at least to me. I don’t expect to know or be capable of comprehending the full extent of God’s plan, but I think some manner of revelation is more plausible. Okay, so where do I see revelation and the guiding hand of God? Well, pretty much the Bible. There’s a lot of comfort in the Bible. There’s also a comforting internal consistency and consistency with natural law that conforms to something being true. The Ten Commandments make a lot of sense. The Golden Rule makes a lot of sense. These are rules I can live by, or at least strive to live by, and hope everyone else does too. A lot of the Bible is backed up by other historical sources. A fact that early Judaism survived and Christianity even exists seems so implausible in light of so many challenges and potential challenges if it was false, as to lend it credence. That the Catholic Church survives, seemingly unchanged in doctrine, seems miraculous. These are the stirrings of faith, in concordance with the promises of the Bible. Belief in infallibility seems to come naturally in light of the historical evidence and all of the above. Where some might see circular reasoning, another might have faith and see divine consistency, a self-evident truth one simply can accept.
I can’t say I know how Bishops know exactly when they’ve got something right, all I know is it seems to take a great deal of study, prayer, and debate, and ultimately they don’t announce anything new very often. That seems about right to me.