What is the relationship between dipole moment and solubility?
It's all about the energetics (assuming an isolated system--happy to elaborate in the comments): as acetonitrile has a large dipole moment, it can readily stabilize the individual ions. See this discussion of solvation energy.

The more subtle case is in the case of nonpolar compounds. In this case, the large dipole can create an induced dipole in the nonpolar molecule. Then the dipole-induced dipole attraction would be larger in magnitude than the spontaneous dipole-spontaneous dipole interactions present in nonpolar-nonpolar interactions (which are also still present as small fluctuations in the polar-nonpolar case). See this discussion of how charges/dipoles/multipoles induce dipoles and then subsequently attractively interact with said dipoles.