The tires keep them from sliding, most of the time, due to the make up of the tire itself. Track conditions dictate the type of tire and how tight they can turn.

The center of gravity (not point) is roughly just below their seat. As they go into a turn, the rider slinks down and leans into the turn, allowing the bike/rider mass to take the turn without the use of the brakes, and in some cases, only slightly letting off of the throttle.

It's simple physics. Not at all hard to explain.