Bitcoin mining: What am I missing about understanding how the nonce value is used?

So I understand more or less how bitcoin mining works, but I'm missing something.

I get that bitcoin miners change the nonce in order to vary the hash. But why are the nonces so low? Before seeing the nonce value displayed on a blockchain explorer, I thought that the nonce was a 256 bit number, and people would just randomly generate values to try. This would obviously never cause collisions, and seemed fairly simple.

But I realise now, that the nonce values are incredibly small (around a billion). So is something changing in the data to mean that nonces are reset quickly enough to not exhaust the nonce range? Is it the timestamp, or something else as well?

I would love to know how this works, please.