Question

Why minimum volume of the solvent needs to be used during crystallisation process?

Answer 1

because you want (as far as possible) to only dissolve the desired product (not the impurities). So the less solvent you use the more likely you are to not dissolve any impurities (which can then be caught in filtration).

It's not a guarantee, of course, because your impurities may well be as soluble (or even more soluble) than your desired product. But it helps