If the bike isn't leaned when you start a turn, centrifugal force (plus the fact that the center of gravity is off the ground) would mean that the bike would fall over to the outside every time. By leaning, roughly speaking you're using the weight of the bike to balance the centrifugal force that wants you to fall to the outside. Therefore, we have to lean the bike to turn without falling (duh!)
There are 2 ways to lean the bike: countersteering or shifting your weight side to side, where the motorcycle shifts the opposite direction.
Think of countersteering as temporarily unbalancing the bike to cause it to lean. The idea is to use the centrifugal force from a short turn in the opposite direction to create the lean necessary to balance the turn. The centrifugal force in turn depends on a) how fast you're going (faster = more force), b) the radius of the turn (tighter = more force), and c) the mass of the bike+rider (not important for this discussion). The size of the centrifugal forces determines how fast you lean over.
What this basically implies is that at very low speeds, it's a little harder to generate a lean from countersteering: because your speed is low, you need a tighter radius to get the same centrifugal force. This is easy to see in practice -- as you go very slowly, say at a stoplight, you need more dramatic steering motions to stay in balance.
Another take on the same idea is that you have slightly less control from countersteering at low speeds, and therefore body positioning gives you relatively more control compared to countersteering at these speeds. That's why counter-weighting is important to making low speed maneuvers like U-Turns.
Here's a video demonstrating most of this:
Link to original page on YouTube.