Why do passengers deviate backward with a sharp increase in the speed of the bus, and for a sudden stop, forward?

Not for the 7th grade, of course.
A sharp increase, and indeed an increase in speed. Arises from acceleration
Try pouring water into a bottle and quickly moving it back and forth.
You will see that it bends down like the passengers on the bus.
The fact is that when braking, acceleration is directed against the movement, and when gaining speed in the direction of movement (not necessarily parallel to the velocity vector).