Maybe it helps to see backgammon as a game of american football (sorry for the real football fans, but yeah, this is what it's called
). I'll ignore (back-)gammons and the cube. The middle line of the game is equity 0, or 50% GameWinningChances. Your scoring line is equity +1, or 100% GWC, Opponent's scoring line is -1 or 0% GWC.
The game starts with the ball in the middle: equity zero (or 50% GameWinningChances). A lucky roll brings you closer to the goalline, as equity increases. An unlucky roll takes you farther away, decreasing equity. The good side of the middle line is positive equity, or >50% GWC; the bad side of the middle line is negative equity, or <50% GWC. The same goes for opponent's rolls, but with the opposite effect for you. So basically, luck comes in two forms for a player: good luck on his own roll, or bad luck on opponent's roll.
If this was all there was to backgammon, just luck and no skill, then it's just waiting until one side gets enough cumulative luck (including bad luck for opponent) to cross the goalline (i.e. win the game). Obviously both sides have equal chances, and the total amount of luck needed to win a game this way is half the playing field, in bg equity +1 (or 50% GWC).
Now, the skill factor basically means that you can make errors in the game, which will bring you farther away from the goalline than just the luck factor would have determined. The same is true for your opponent, but with the reverse effect for you: an error by opponent will bring you closer to your goalline, without needing luck.
An example: you win the opening roll 3-1. This is lucky. You go from the middle line to the next line, ten yards closer to the goalline, just on luck alone. Equity wise, say you jump from 0 to +0.1 and in GWC from 50% to 55%.
Now suppose you play the 3-1 wrong, 24/23 13/10. This is like a 0.1 error, or -5% GWC. It will set you back 10 yards, and you're back to the middleline.
You were lucky, but you haven't made any progress in the game, because of your blunder. Basically, you threw away some of the luck you got, by playing a bad move. If you do this often, and your opponent doesn't, it's easy to see that you will need more good luck to win a game, than your opponent does!
So, when a good player plays a bad player, the latter will make a lot more errors, bringing the better player closer to the goallline all the time, without him needing luck for that.
As it is however, single backgammon games tend to have a huge luck factor, meaning that the advantage a better player creates by the above process is typically quite small. Like, with equal luck for both sides, after a series of moves, the better player has moved 10 yards from the middle line, closer to the goalline. Still plenty of yards to go, and luck will be needed to cross that distance. But your opponent will need even more luck to win this game, as he's farther away from his goalline.
In equity terms: after a bunch of moves in a game, the better player can expect to have gained f.i. +0.2 equity due to opponent's errors being worse than his. Now the better player needs "only" 0.8 luck to win the game; the worse player needs 1.2 luck to win the game. 0.8 luck is more likely to happen than 1.2 luck. So yes, the better player gets lucky more often
, but by smaller amounts
; the worse players gets lucky less often, but by bigger amounts (as he needs them!). Multiply the two (frequency of luck * amount of luck) and the total luck both sides receive is still equal.