Because your arm is at an angle to the rows of the keys (with a non-split keyboard), so when you extend a finger, it doesn't go straight up. With staggered keys, you can reach the key on a row up by only extending your finger (i.e, moving it in line with the direction of your underarm), without having to move any other fingers aside.
This argument may justify that staggered keys are better for one hand and much worse for the other hand. Your hands are symmetric, why would you want an asymmetric keyboard (split or not)?
How do you mean? For both the left and right hand, you extend the fingers, pretty much symmetrically. It's not perfectly straight depending on how the staggering is done, but straight up is worse than all the alternatives.