Grab your cocoa.
When you make a (left) corner at speed, you, actually turn the handle bars right. The forces involve want to tip the bike left, so you get your lean.
If you don't lean against the forces, the bike will tip over to the outside of the corner, if you lean too much, your bike will fall to the inside of the corner (ever chopped the throttle mid turn?).
Making theoretical sense so far? Good, I'll continue.
So between tipping over and falling in is the optimum range for the angle of lean, conveniently the centre of gravity. The CoG extends as a line to the contact with the tarmac. It is the sum of you and the bike. The CoG is higher with you on the bike than just the bike alone.
Now, bike stability. Bikes are most stable upright, and the less lean in a corner, the more stable the bike must be. You can influence the bike lean by moving the centre of gravity. How? By leaning over the bike in to the corner, or conversely, you can lean yourself to the outside (ever had a pillion try to lean out when cornering?).
So by hanging off the bike to the inside of a corner actually allows you to keep the bike more upright than if you kept yourself in line with the bike. You can also lean the bike more by leaning out. Try pushing your left handle bar away from you, the bike will tip plenty.
On a sportsbike by leaning forwards and in to the turn (hanging off the bike) you are keeping the CoG low, so fewer forces try to throw you to the outside, and you reduce the bike lean, increasing relative stability. Sticking your knee out marginally moves the CoG further, and if leaning enough you can get your knee/elbow/helmet on the ground.
So getting your knee down is just cornering with added bike stability.
Sleepy? Thought so. Have a good kip.