The main observations focus on broken symmetry. Polynomial-time algorithms often have to start by choosing some unit of work, but without a requirement that work units be ordered; therefore some choice is required in order to pick a starting work unit. For example, to process a graph, one usually must start by choosing an edge or vertex without limitation.
The claim is, effectively, that SAT requires significant choices to be made in which direction to compute in order to solve its instances in polynomial time.
I confess that, although I've read the proof, I neither am convinced by it nor can find problems with it. I need to study it more.
I'm not sure how the barriers are handled. The relativization barrier, I can imagine, is handled by showing that abstract state machines with oracular advice can turn seemingly-insignificant choices into significant choices, so that the proof doesn't relativize. I don't know about algebrization, though.
In terms of Impagliazzo's five worlds, this result would indicate that we are in Minicrypt or Cryptomania, as is commonly believed already.