As the title to Mumford's famous paper "Toward an enumerative geometry..." suggests, knowing the cohomology / cycle theory of the moduli space of curves allows one to answer enumerative geometry questions for curves.

Here is an (very concrete) example that came up in real life for a student of mine. He had a family of genus 2 curves over $\mathbb{P}^2$ constructed by taking the family of lines in $\mathbb{P}^2$ and for each line taking the double cover of the line branched at the six points given by the intersection of the line with a fixed (generic) sextic curve (equivalently, this is the defining linear system on a generic genus 2 $K3$ surface). He needed to know how many of those genus two curves admit a degree $d$ map to a fixed elliptic curve $E$. This is a calculation in the cohomology ring of $\overline{M}_2$: we are looking for the number of points in the intersection of the two cycles in $\overline{M}_2$ given by the two families of curves: the 2-cycle given by the family over $\mathbb{P}^2$ and the 1-cycle given by the locus of genus 2 curves admitting a degree $d$ map to $E$ (since the dimension of $\overline{M}_2$ is 3, we expect these cycles to intersect in points). This intersection problem is dual to a cup product computation in the cohomology ring of $\overline{M}_2$. Using Mumford's complete description of this cohomology ring, one can determine how to express each of our cycles in terms of Mumford's generators and then compute the cup product. There are some minor issues to fuss with involving the orbifold structure and some general position issues, but the problem is essentially solved with the cohomology of the moduli space.

This was a rather specific example, but maybe you get the idea: if we want to count the "number of curves satisfying this that and the other", we try to formulate the question as a problem in the cohomology ring of the moduli space.

As the title to Mumford's famous paper "Toward an enumerative geometry..." suggests, knowing the cohomology / cycle theory of the moduli space of curves allows one to answer enumerative geometry questions for curves.

Here is an (very concrete) example that came up in real life for a student of mine. He had a family of genus 2 curves over $\mathbb{P}^2$ constructed by taking the family of lines in $\mathbb{P}^2$ and for each line taking the double cover of the line branched at the six points given by the intersection of the line with a fixed (generic) sextic curve (equivalently, this is the defining linear system on a generic genus 2 $K3$ surface). He needed to know how many of those genus two curves admit a degree $d$ map to a fixed elliptic curve $E$. This is a calculation in the cohomology ring of $\overline{M}_2$: we are looking for the number of points in the intersection of the two cycles in $\overline{M}_2$ given by the two families of curves: the 2-cycle given by the family over $\mathbb{P}^2$ and the 1-cycle given by the locus of genus 2 curves admitting a degree $d$ map to $E$ (since the dimension of $\overline{M}_2$ is 3, we expect these cycles to intersect in points). This intersection problem is dual to a cup product computation in the cohomology ring of $\overline{M}_2$. Using Mumford's complete description of this cohomology ring, one can determine how to express each of our cycles in terms of Mumford's generators and then compute the cup product. There are some minor issues to fuss with involving the orbifold structure and some general position issues, but the problem is essentially solved with the cohomology of the moduli space.

This was a rather specific example, but maybe you get the general idea: if we want to count the "number of curves satisfying this that and the other", we try to formulate the question as a problem in the cohomology ring of the moduli space.