I must say that the various proofs of this theorem are really difficult to absorb. I'll probably have more than a few questions about them. Meanwhile, I found a SUNY-Stony Brook website that seemed like a useful reference, but it includes the following statement which I think says exactly the opposite of what it should say:

Isn't that backwards? To be precise, we must show that every problem in NP is polynomial time reducible to SAT. So, to paraphrase that in a more colloquial way, wouldn't it be more accurate to say that we must show that SAT is at least as hard as any problem in NP?