Think of it this way. If you have a solved cube and you only do U, D, L, and R moves to scramble it, then the cube can be solved even using only U, D, L, and R moves.
Hold a solved cube with green on front and white on top. Look at the the white/orange edge at UL. It's solved, so clearly it's correctly oriented. Now do as many U, D, L, and R face turns as you want to as a scramble. Really scramble it up a lot using only those faces, and then find the white/orange edge, and using only U, D, L, and R face turns, move it back to UL. See that it is still oriented correctly.
Now, with the white/orange edge solved in UL, do L F U, and see that it is now back where it was, but incorrectly oriented. In fact, not only is that edge incorrectly oriented, so are all the other edges that were on the F face when you turned it.
So, when you do a quarter turn on the front or back face, you are "toggling" the orientation of all edges on that face. The ones that were good become bad, and the ones that were bad become good. This is also why you can do a half turn on F or B and not have changed edge orientation, since a half turn is just two quarter turns, and so you are toggling the same edges twice to no overall effect.
Now with wide and slice moves, you just need to be aware that you are changing the orientation of the cube by moving the centers. So for example, if you scramble a cube using U, D, L, and R face turns, then do u', the center that was on the left is now on the front, and the front is on the right. You essentially just did a D' and then a y'. So, move set would change to U, D, F, and B, which are really the same faces as before, you're just looking at the cube from a different side. Same with slices, in fact EO in Roux takes advantage of slice turns to do EO toggles of the top edges by moving the front or back center to the top and then doing U. For example, if you have four bad edges at UL, UR, UF, and DF, you can do L R' F, which brings the UL and UR edges down to the front face and toggles them all to good, or you can do M' U which brings the UF and DF edges to the top along with the front center and toggles them all to good.