Because at the same time as pushing at the pins, you're applying turning force to the lock (that's what the black thing at the bottom is for - called a torque wrench). That turning force is resisted by the pins where the break (between red and blue) isn't yet aligned with the break between the cylinder and the stationary part of the lock.

Because the machining of the lock can never be perfect, only one of the pins will be providing the majority of that force (shown in this gif by a little grey oval). When you push up that pin, it catches on the lip between the clyinder and the stationary part and stays up. The cylinder rotates a tiny bit, and then is stopped by the next pin (see the grey oval move). Repeat for each pin, and the lock opens.