Can anyone please explain why Kasparov would not have played 5. axb4 here? 

looking for a forcing line that makes taking the bishop truly deadly 

This is a mistake in notation most likely. There is nothing wrong with axb4. It wins immediately.

There you go!

Kasparov was seeing if his opponent would notice the pawn attacking the bishop, so he doesn't take that bishop on move 5. After about half an hour or how ever long it takes Kasparov to get all the way back round to Nolsoe Eydun he realised his bishop was under threat! So he traded it off.

if you look at the whole game, youll see a lot of other moves which have no explanation as well, like move 31. the game is probably false and was never played.

yeah it's faulty notation or someone messing around. There's no reason for most of these moves. Don't sweat it.

It's called the Larry's Gambit Declined.

I don't know Fonix, but "he missed it" doesn't seem plausible. Go on to move 19. Why wouldn't he take the rk with his pn?

It's definetly a misprint. Here is the real score.

That game was full of nonsensical moves. The queens went off at different times, unforced. The rook takes a knight, and then takes the rook which could have taken it no problem.

Either this is a false game, mistaken notation, or it was 10 second blitz:)

Danke! AnthonyCG

Thanks AnthonyCG