The following answer is unfortunately not quite correct, but it may be useful anyway. I will of course be ignoring any virtual fundamental class issues.
Imagine that you are computing a Gromov-Witten invariant where you require the i-th marked point to land at a specific point (i.e. your i-th insertion γi is the class of a point), and now lets add aditionally the i-th psi-class as an insertion. You can restrict to the subspace of maps with $f(x_i) = x$ for some generic choice of $x \in X$. Fixing an arbitrary non-trivial map $\Phi \colon T_x \to k$ gives you by composition a map from the relative tangent bundle of the universal curve over $M_{g, n}(X)$ at the section xi to the trivial line bundle, in other words a section $\phi$ of the relative cotangent bundle of the universal curve. It will vanish on curves which are tangent to a hypersurface through x with tangent direction matching the zero-locus of the map $\Phi$.
So you can think of Gromov-Witten invariants with psi-classes as counting maps which additionally satisfy tangency conditions at the marked points.
Why is this not correct? The zero locus of $\phi$ computes the Chern class of the relative cotangent bundle at $x_i$ over Mg, n(X), which is not the same as the pull-back of the $\psi$-class from Mg, n. Insertions of the former are sometimes called "gravitational ancestors", and the difference to gravitational descendants is described explicitly in alg-geom/9708024.