When we start from a steady-state balanced & non-turning condition (see disclaimer), and we shift our weight to one side, the bike naturally has to lean in the opposite direction. The bike's leaning is required to balance the force required to shift your weight.
When a bike is leaned, however slightly, the combined effects of steering angle, rake, trail and tire profile - cause the front forks to steer into the bike's direction of lean (see note). In motion, this then causes the front tire to out-track in the direction of bike lean, moving the bike further out of balance relative to the rider and the CoG further into the direction of the body lean.
At which point the bike will roll past vertical into a lean in the direction of the body lean & combined CoG. Then the bike will turn. So while it seams that all you are doing is shifting your weight, in actuality the bike naturally counter-steers in response to the weight shift.
Discussions about countersteering are about using this natural effect of bike dynamics to consciously control the bike with greater precision and greater urgency on command as needed. Those people that have learned & mastered the technique enhance their ability to control their bike, especially in critical conditions. It's about learning as much as we can about how bikes work so we can ride more safely in a range of conditions.
Disclaimer: A motorcycle in motion is never in a steady-state of balance. In motion all single-track vehicles lean and weave right and left continuously. The effect is most pronounced at slow speeds but even at speed it happens continuously. There's no way to stop it since we are balancing over two inline contact patches.
Note: You can see this effect by holding a bicycle from the top bar while vertical w/ neutral steering angle. Now lean the bike slightly in one direction or the other. Without any input at the handlebar, the front wheel will turn into the direction of lean. Lean the bike in the other direction and the front wheel will turn towards the new lean angle.