Why must we filter the solution in a recrystallization experiment?
Because your solution will contain contaminants: dead skin; fly poo; hairs; little bits of fluff and other insoluble material. Every recrystallization performed purifies the material.
Crystallization is a very important method of purification. Look at salt crystals in the salt cellar, or sugar crystals in the sugar bowl. All of these crystals have undergone large scale recrystallization before getting to the table.
By the way, can you tell me what a saturated solution is? If you say that "the solution holds all the solute that it can" you will cause your chemistry teacher to cry!