This is a little confusing because he does not appear to be using "page" to mean a memory page. So this is not really a 6502 question as much as it is a question about this specific game.
From what I can tell having glanced at the code, I would bet that the level maps are broken into sections (which he's calling pages), and when the player transitions to a new page, new stuff on the map gets loaded.
I deduced this specifically from some code starting at line 5997:
CSetFDir: lda Player_X_Position
clc ;add or subtract from player's horizontal position
adc ClimbAdderLow,x ;using value here as adder and X as offset
lda Player_PageLoc ;add or subtract carry or borrow using value here
adc ClimbAdderHigh,x ;from the player's page location
Player X position seems to be tied directly to player Page location. As you move further right, you move onto a different page.
This also suggests that pages are 256 ?pixels? wide... so basically one "screen" worth of map?
Keep in mind this is all kind of a guess, as I looked at this for less than a minute. But I can say for certain that "page" in this context is referring to something specific to this game and not to anything related to 6502 or NES architecture.