This term has an entirely different meaning in backgammon, compared to the basic sense of the word. In backgammon, to button up means to make safe a single checker by bringing it together with another checker.

A single checker is called a blot, and it is very vulnerable, so by buttoning up with another checker, you will make its position safe. It is closely connected with another term: to book a checker, the only difference between the two is that when you book a checker, you also make a point, not only make it safe.