"legitimate interest" is a legal term with specific definitions in the GDPR. (And indeed it refers to the interest of the
site, not yours)
IANAL, but as I understand, it refers to data collection that is inherently needed to perform a service.
E.g., a pizza delivery service has a legitimate interest to know the address of the place where it should deliver the pizza to - because, well, otherwise they can't deliver the pizza.
In such a case, the GDPR wouldn't require the pizza place to get consent. (the GDPR requires that a service is performed even if consent is denied, so without the legitimate interest exception, the pizza place could end up in a legal catch-22 if someone ordered a pizza but denied consent to collect the address.)
The basic idea seems perfectly reasonable to me, but of course sites always tried to stretch the "legitimate interest" definition as wide as they could get away with, and this seems to be the latest iteration of that.
I have no idea where the latest fad of claiming all kinds of ridiculous things as legitimate interest as long as there is an "object" button comes from, but I imagine there was some court case that decided this was borderline legal. If anyone else knows more about this, I'd really like to know as well.
But at least I think this is why many consent popups ask the exact same questions twice, once as "consent", off by default and once as "legitimate interest", on by default: They are simply trying their luck on two separate legal avenues. (Not that this would make any sense from a UX point of view or from the intent of the law. But I guess it does make sense from a "scummy lawyer" point of view)