Why can he see into different playthroughs of the game? How does he know more than he should?

Well, many theories on this one. Among the most popular is that he's seen data from time-based research experiments with W.D. Gaster and has seen the 'timelines stopping and starting' on the graph. Another theory is that he has a Time-proof box that he uses to keep information between resets. Yet another theory is that he has full retention of memories through timelines for unknown reasons. Another is that all characters retain some memories, but only Sans knows that they really are memories and not just dreams. I personally like the first and second theories the best.

sans is sans, that's my theory.

I think I saw something once that said that he remembered the past timelines better due to being asleep at the time a reset would send the character back. There was also the theory that most people can remember it (ex: Toriel remembering if you chose a pie flavor from your first run) but just subconsciously forget it/block it out. However, Sans is somehow able to remember the timelines itself to an extent, whether it's due to his more grim and mature mind or just by some magical or sciency reason (probably either being a really powerful monster or exposure to things from Gaster's expieriments.)

Welp, that's all I got that hasn't been fully explained.