Explain the laws of physics involved in Watney's final launch in The Martian and the connection to the rescue team.

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There are a lot of physical laws at work concerning the launch of Watney's rocket to join up with the rest of his crew, and even though the book is at explaining science topics, the book simply can't go into every physics concept of rocketry. For the purpose of this answer, I'll stick with Newton's three laws of motion.

The third law states that, for every force exerted, there is an equal force exerted in the opposite direction. As the fuel is ignited and vented out the bottom of the rocket, there is an equal force being applied in the upward direction. This force is what will eventually cause the rocket to accelerate out of the Martian atmosphere. Newton's second law applies the moment the fuel is ignited. Acceleration is force divided by mass. Watney can't change the force of propulsion, but the rocket can be accelerated due to lost mass. The lost mass is a result of the fuel being ignited and vented out of the spacecraft. As the craft loses mass, it accelerates more and more.

Once out of the Martian atmosphere, Newton's first law takes over. Watney's craft is moving at a particular velocity. The same is true for the rescue vehicle. Those velocities won't change unless an unbalanced force acts on either craft. This is what the crew of the Hermes is forced to do with their controlled explosion that changes the velocity and orientation of the ship. Fortunately, all goes according to plan, and Watney is saved.

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