The way I think about it is in terms of interior points. If A is an open subset of the domain, then every point x in A is "insulated" by a small neighborhood of other points in A. If f is an open mapping, then the same is true for f(x): it is insulated by a small neighborhood of points in f(A).

For example, f(x)=x^2 isn't an open mapping of R -> R: A = (-1, 1) is an open set, but f(A) is not, since f(0) becomes stranded on the boundary of f(A) = [0, 1).