Question

Why do you fall in the forward direction when a moving bus brakes to a stop and and fall backward when it accelerates from rest?

Answer 1

because of inertia.Intertia is a property due to which a body opposes to change its state. When the bus is moving we are also moving with the same speed as bus and when suddenly bus stops we are still moving due to inertia . That's why we experience a fall in the forward direction

Answer 2

When the bus it at rest your body is also at rest. As soon as it starts it gets into motion while your body is still at rest. This causes your body to fall backwards. Similarly both your body and the bus are in motion and when bus stops suddenly, you body is still in motion hence falls in forward direction. 
Consider the first action similar to someone pulling a static dog from its leash and the second one is similar to someone throwing a water balloon at a static wall.