The reasons for caring which occur to me seem to fall into two broad categories.

First, line bundles are useful for identifying interesting substacks of your moduli stack. You usually first encounter this idea in GIT theory, but the philosophy applies more generally. See, for example, Teleman's papers on the stack of G-bundles on a curve, and Jarod Alper's paper on "good" moduli spaces ( http://math.columbia.edu/~jarod/good_moduli_spaces.pdf ).

Second, sections of line bundles are a good substitute for/generalization of functions. For example, on the moduli stack of curves, the only holomorphic functions are constant, but there are plenty of meromorphic functions, which are naturally thought of as sections of line bundles. This point of view shows up, for example, in the theory of modular forms (where it clarifies their transformations under SL(2,z)) and in the Beilinson-Bernstein approach to representation theory (where you use "twisted D-modules", which act on sections of a line bundle on the flag variety rather than on functions, to get representations with non-trivial central character).

Is that roughly what you were wanting?