In order for molecules to precipitate out of solution, they need to aggregate together. Amino acids that have zero net charge can aggregate together much more easily than those that are charged.
Molecules that have net charge need counterions to aggregate with them to offset the charges or the electrostatic repulsion will be too great.
The concept of isoelectric point is more often applied to polypeptides and proteins than to individual amino acids. In those much larger molecules, the tendency to aggregate when there is no net charge can be quite high. This is especially true if there are any hydrophobic regions that can favorably associate with each other and reduce the interaction with water.
It is important to note, as mentioned in the comments, that this does not mean that all amino acids and proteins are insoluble at their isoelectric point, simply that they tend to be less soluble at that pH than at pH values where they carry net charge.