How are problems classified in Complexity Theory? I'm reading Sipser's Introduction to the Theory of Computation (3rd edition). In chapter 0 (pg. 2), he says we don't know the answer to "what makes some problems computationally hard and others easy," however, he then states that "researchers have
discovered an elegant scheme for classifying problems according to their computational difficulty. Using this scheme, we can demonstrate
a method for giving evidence that certain problems are computationally hard,
even if we are unable to prove that they are."

So my question is: HOW is it possible to classify problems according to their computational difficulty, if we don't even know what makes a problem computationally easy/hard in the first place?

Also, what/where is this "scheme" that does this classifying. (I did some googling and couldn't find anything)
That's what you get when you distill a whole bunch of theory to a wider audience.

In his book, Sipser addresses a general audience at the undergraduate level, possibly with no notion of computability theory; hence, he can only hint at concepts which are to be given a more formal treatment later on in the book. The part you cite is from chapter 0 (i.e., not really a chapter), whereas the material for complexity theory only appears at the end (i.e., part three). This is why the passage is so fuzzy. Most likely it is intended only as motivation and to give a broad overview for the topics to be covered in the book.

The "scheme" Sipser is talking about are reductions. If we know a problem $A$ is reducible to a problem $B$, then we know $B$ is at least as hard as $A$. (Incidentally, this is also why it is common practice to denote reductions with a "$\le$" sign.) This gives us a way of ordering problems according to their difficulty, at least for those having reductions we are aware of. As Sipser states, though, by using only reductions "we are unable to prove" whether the problems are really hard or not; reductions only give us relative, not absolute notions of hardness. This is why separation results are still rare in modern complexity theory: We have a bunch of reduction (e.g., NP-completeness) results, but only a handful of separation results (e.g., the time and space hierarchy theorems).