How can a car's engine move the car?

The car's engine tries to make the wheels turn. However, the wheels encounter friction against the road so they cannot just spin. As the road has much higher inertia than the car, it will not move when the wheels want to turn. Instead, it is the car that moves.

The end effect is that the engine pushes against the road, just as you do when you push the car: your feet are on the road, allowing you to push the car forward. In the case of the car, the wheels are on the road and they can push the car.

Pushing the car assumes you have enough force, and the road is not too slippery. This also applies to the engine: if it does not have enough power, or the road is too slippery (icy), it cannot push the car.