I would also like to know why when I'm in a tight turn and feel like I'm running out of room, rather than braking, which seems to exacerbate the feeling of loss of control, the thing to do is to accelerate and the bike seem suddenly more stable. My buddy says the rear wheel 'bites' into the road and traction improves. Not sure this is true though I can definitely 'feel' an increased bite.
Everyone has given great answers and the Keith Code books are great, the MSF Riding Course is a must for all riders as they cover the "press right (which is to say steer left) go right" countersteering.
To address your question it was explained to me as an issue of forces on a gyroscope. They always effect it at a 90 degree angle. So braking forces tend to stand the bike up. IF you absolutely have to brake in a turn, you must input more countersteering to simply maintain the same line. Usually a conscious effort as these things are not intuitive, takes training and practice. As many have said accelerating forces tend to steer the bike down into the turn, or at least stabilize it. BUT, the three forces of braking, countersteering, and centrifugal will rapidly eat up available traction and down you go. Better to stay off the brakes and use available traction in countersteering. You don't have much choice on the centrifugal.
Also aggressive countersteering will scrub off a lot of speed without the negative "standup" gyroscope reponse that braking gives. Win Win In Keith Code's book he says you have to practice staying off the brakes because the act of throwing the bike into a turn with aggressive countersteering will slow you so much, if you brake as well all of a sudden you're going too slow.
If you are too hot in the turn more often than not do not decelerate or brake, as this will stand the bike up, the physics of force on a gyroscope. hold steady or input a little, this will maintain or lean the bike more and ... as a racer once told me...trust your tires.
Have you ever held a bicycle wheel out in front of you and have someone spin it. Then input some force into the axle and the wheel will respond 90% out. Its really weird.
Other important points are "target fixation". Your bike WILL GO where you look. If you're too hot in the turn and you look to the outside that's where you will end up. If you look down and out the inside of the turn, you'll stay on line faaster than you thought possible. I once read Kenny Roberts teaching how to flat track: "You look at the hay bales, you'll be in the hay bales". I was a lot better rider after reading that.
Soooo many tragic crashes occur from lack of knowledge and training, not because it was a particularly dangerous situation.
Then, sometimes we just fall down.
