Why are aprotic solvents unable to solvate negative ions well?
This can be attributed to the lewis acid/base properties of the solute/solvent. Anions (negative ions) and aprotic solvents both function as lewis bases. They 'repel' each other. However, a cation (positive ion) functions as a lewis acid and is 'attracted', resulting in greater dissolution.
How well a compound dissolves in a given solvent depends on a multitude of factors. This includes the two very important ones discusses here, polarity and lewis acid/base character.