In a typical keyboard+mouse FPS game, for example, you get 2D analog "look" controls with the mouse, but much more limited 2D "move" controls with the keyboard.
With two analog sticks/trackpads/mice/etc. you can get the same range of control over both. That's what the two "planes" give you.
In most FPSes your move commands are binary, "W" or nothing. You can alter them a bit with combinations, Shift-W to sprint or perhaps Ctrl-W to crawl but you're still limited to a handful of preset speeds.
I could easily imagine being able to use a continuous axis to encapsulate walk/jog/sprint being quite a bit more intuitive and precise.
What about circle strafing a moving target while jumping/ducking and shooting? For me, that takes 3 simultaneous inputs on my left hand and 3 on my right hand. On each hand, some of those inputs vary or alternate. I can't speak for "most FPS" but competitive FPS require this coordinated action.
With two analog sticks/trackpads/mice/etc. you can get the same range of control over both. That's what the two "planes" give you.