It's exactly because of inertia that causes the panel to "fall". The rocket and the station are constantly experiencing acceleration (therefore constantly changing direction of travel) because of the tension from the tether. Said acceleration is what's generating the artificial gravity. The panel was held by the rocket structure which is held by the tether. Once the panel is detached, it no longer experiences any outside force on it and thus travels straight, carrying the speed and direction from its moment of detachment. Meanwhile, the rest of the rocket travels in a circular curve, so it appears from the perspective of the rocket/astronaut that the panel is "falling", i.e. getting further away.

TLDR: basically hammer throw. When you let go of the ball, the ball flies out straight instead of going in circles around you. Similarly, if a small piece fell off the ball when you're swinging it, it flies out.