Its main use is to define the cohomology of an algebraic stack with coefficients in, say, the étale sheaf
To understand the problem that motivates the notion, consider the classifying stack
in the étale topology;[1] i.e., just a point.
However, we expect the "correct" cohomology ring of
as the ring should classify line bundles.
should be defined using smooth topology for formulae like Behrend's fixed point formula to hold.