As I remember, it not that the particles physically spin (at least no one has experimentally observed them spinning, the particles are too small to see) but that they can carry angular momentum. Large objects that we are familiar with that carry angular momentum are spinning, so 'spin' was used as an term, but I've heard people say to think this way is misleading.

Almost everyone's first question in a Quantum Mechanics class is 'why do things behave this way?' As far as I have heard, no one (even those that developed the equations) has anything better than 'these equations produce results consistent with experiment'.