What in the world does this formula mean?

This is the famous Kazhdan-Lusztig character formula. It expresses the formal character of an irreducible highest weight representation

as a sum of characters of Verma modules

, with certain integer coefficients

. These coefficients are found by first computing a recursively defined family of polynomials

, where q is the variable and y,w range over the Weyl group. These polynomials are computed by studying the Euler characteristic of the intersection cohomology complex of the Bruhat cell attached to w along various other Bruhat cells. Once the polynomials are known, we evaluate them at q=1. The coefficients of these polynomials encode the singularity of the closure of the Bruhat cell attached to w.