How a Motorcycle Steers
When a motorcycle is leaning in a turn, the inside edge of the contact patch has a smaller rolling diameter than the outside - like an icecream cone laying on its side.
The cone wants to roll in a circle into the turn. This causes a twisting force in the tire which steers it into the turn independant of the wheel (referred to as tire drift). Until the wheel catches up, the tire is actually steering in a slightly different direction than the wheel.
Trail braking will increase the size of the contact patch and increase this twisting force, steering the wheel to the inside of the turn. The more braking thats use, the more the bike will feel like it's trying to stand up. This will require more force from the rider to keep the desired line through the turn.
Release the brakes and the effect fades.
Many other factors can affect this sensation. Different tire profiles will affect the tendency of the contact patch to steer into the turn. A sharp Vee profile front slick will steer very differently than a rounded touring tire.
To make things even more complicated, pneumatic trail can change the steering effects of the tire based on the air pressure in the tire and the amount of front brake being used. The lower the air pressure the greater this change will be, making the bike feel "loose".
You can see pneumatic trail and tire drift at work in this picture of Nicky Haydens front contact patch:
Next time you're riding keep in mind that turning the bars into the turn causes the bike to stand up, while countersteering causes the bike to drop into the turn.
Understanding the forces at work can help explain what you're feeling while you experiment with things like trail braking and adjusting your line mid-corner.
If you want to learn more about these concepts, take a look at the excellent book "Motorcycle Design and Technology" by Gaetano Cocco and "Top Dead Center" Collections 1 & 2 by Kevin Cameron. Both are full of brilliant explanations of exactly how a motorcycle does what it does.
Here is a full diagram of the forces at work while we are having fun in a turn.