Nothingness: the doubling cube only makes sense in a match of more than 1 point

by declining the cube offer you will lose that game, but not the match (unless if its the last point your opponent needs)

in the end game when you are far behind your opponent, your opponent can offer the double, you will of course decline, and the game will finish a bit faster :)

in the end game when both are not that far apart, one can offer a double if he think he will win .. if he was correct then he wins double the points he would have otherwise won

the trick is to find where you will still win, and your opponent might still accept .. i you can notice that position, then you win double the points your opponent might win in the same kind of game, and you might win the match although your opponent might be of equal strength in backgammon without the cube

of course there is a bit more as alan explained : when the doubling cube is available you can also win gammons and backgammons .. a gammon (no men beared off by your opponent) is worth 2 points, without any doubling applied yet, a backgammon (opponent still has a men in the starting quarter or on the bar) is worth 3 points ... so a game which was doubled once and became a backgammon .. is worth 6 points :)

now the trick is to find out .. when you are winning a game .. wether you can turn it into a gammon (which will give you 2 points without doubling) or if you wont be able to do so .. if you double and your opponent declines you will only get 1 point .. but it is a certain point

the best way to learn it is to try it :)