To understand solvation, the first and best approximation is to look at the definition of the term. The IUPAC says about solvation:
Solvation
Any stabilizing interaction of a solute (or solute moiety) and the solvent or a similar interaction of solvent with groups of an insoluble material (i.e. the ionic groups of an ion-exchange resin). Such interactions generally involve electrostatic forces and van der Waals forces, as well as chemically more specific effects such as hydrogen bond formation.
Just to give an opposite example, water interact with anions mostly by hydrogen bonds. On the other hand, aprotic solvent interacts as well with anions but by a less energetic interaction such as van der Waals. Another angle is by looking at the bond energies. Anions strongly interacts with water and weakly with polar organic solvents.
In fact, aprotic solvents do solvate anions. But, the bond energy of that interaction is too small. e.g. van der Waals interactions (third slide)
To support all statements given here, I recommend the following paper:
Solvation of negative ions by protic and aprotic solvents. Gas-phase solvation of halide ions by acetonitrile and water molecules
R. Yamdagni and P. Kebarle. JACS. 1972, 94(9), 2940. DOI: 10.1021/ja00764a007