because the injection goes into the muscle. It causes a bit of injury, which needs to heal. The tenderness can also be associated to the stuff that is in the shot.
Agreed, it goes into the muscle and damages it a bit, so it'll be sore for a few days afterwords. (Also the fluid in there can burn your tissue as well)
It's possible to have an allergy to the preservatives in the shot. It makes your arm sore.