A space probe may be carried by a rocket into space what keeps the probe going after the rocket no longer push?

Inertia. An object in motion tends to stay in motion unless acted on by unbalanced forces. In the earth's atmosphere, the air density causes drag on flying objects. In the vacuum of space, there is no drag because there is no air. A moving object will continue to move in this vacuum indefinitely unless something else effects it - like running into something, or being caught by a much larger objects gravitational pull.