For example, in Moto GP, you see Toseland, Stoner etc stay on their bike despite being near horizontal whilst going round a corner on the track

centripetal force? they're going so fast gravity does not have time to pull them down.

It is a question of physics. I'd need a whiteboard to show you.

But if they did not lay down to the inside, they flip off to the outside!

friction of rubber on tarmac!

The first thing to understand is the bike wants to stay upright. the design is for the wheels to act like gyroscopes.
the rider has to physically pull the bike over. this is all down to technique, & is where the main differences come from between riders.
then we come to the grip of the tyre
when u think of the down force pushing on a bike sideways, on a piece of rubber a few centimetres across. (the usual description is a credit card size)the grip is phenomenal. they r leaning over at an angle of 72 degrees.
if u think 90 degrees is flat. & 45 degrees is quick. u get some idea of the skill & technology involved.

good tires and forward motion.

Great answer Proc. You sure know your stuff. l sometimes wonder how the riders stay on their bikes even doing mx when they rail a berm and the bike is over so much. Skill, Skill Skill.

l say 10points to PROCK! For more in depth info read TWIST OF THE WRIST.