What happens inside when you twist your knee?

A twisted knee is another way of expressing a sprain in the knee. Sprains being when you extend a tendon or a ligament beyond its normal limit, thus causing pain.

Let's take, for example, your Medial Collateral Ligament. Runs down the inside of your leg connecting your Femur to your Tibia. If it is strained, it is pushed beyond its natural ROM (Range of Motion) and micro-tears form on it, depending on the force of the trauma, other ligaments/tendons may or may not be affected.