Skill in backgammon comes in a couple of varieties.
The first is checker moves. You roll the dice, and now have several moves to choose from. You are typically trying to do a bunch of things -- hit your opponent's blots, build points to block him in, and prevent him from doing those things to you on his next turn. About half the time, one play will be obviously right. The rest of the time, you'll have several choices and the right play will not be obvious at all. The stronger the player, the better the choices he will make.
The second source of skill is the doubling cube. If the cube is in the middle or if you own it, you can double, or not, on your turn. Knowing when to double is a skill very analogous to knowing when to bet in poker, except that the bluff element is pretty much gone. In general, you want to double when you're about a 70% favorite in the game, so you have to be able to look at backgammon positions and say things like "Oh, I'm only about 66% here, I probably don't have a double yet", or "Wow, my chances look like 72% or 73%, I'm doubling." Being able to look at a random position and get the percentages correct to within a percent or two (which good players do pretty easily) is tougher than it looks. Then, when you are offered a double, you have to know whether you should take it or not.
Match a beginner (who thinks the game is all luck) against a top player, with the cube in play, and the beginner gets slaughtered, losing 70-75% of the games and almost all the points (since the beginner wins games with the cube at 1 or 2, while the good players wins with the cube on 4 or 8.)
Still don't believe it? Download the free version of Jellyfish (one of the top-rated backgammon programs, on
www.jellyfish.com), set the skill level to 7 (the highest) and play a few games. Watch what happens.