One real life relationship to a water bottle rocket is an actual rocket. It works on the same physics principles. Most notably, the water bottle rocket and a real rocket both clearly illustrate Newton's three laws of motion.
Newton's first law states that the rockets will remain on the launch pad until an unbalanced force is applied. Released pressure in the bottle or ignited fuel provide that unbalanced force.
Newton's 3rd law explains why the rocket moves up. As the fuel source (water or rocket fuel) is vented down and out of the rocket, an equal and opposite force is applied up against the inside of the rocket. The fuel can exit the bottom but not the top, so instead of being vented out the top, it pushes up. The rocket begins to rise.
Newton's second law explains why the rocket accelerates. Acceleration is dependent on the force applied and the mass of the object. The force being applied is constant in rocketry, so in order to accelerate mass must be lost. As water or spent rocket fuel is ejected out of the bottle/rocket, the entire system is getting less massive. With the constant force and mass being reduced, the rocket continues to accelerate until all fuel is used.