What physics is involved for a passenger feeling pushed backward into the seat of an airplane when it accelerates along the runway during takeoff?

It's due to Inertia. The law of Inertia states that a body will try to retain its movement state if possible. This means, if a body is at rest, it will try to remain at rest using forces. Similarly, if it's in movement, it will try to remain in movement.

In this example, the plane is moving forwards, and so do the seats, which in turn push the passenger forwards. The force felt is Inertia acting on the body, trying to counter the movement.

Newton's laws of motion, #1 and #2, explain everything the passenger feels in that situation.