How can gas pressure be defined at a molecular level?
This is a question which is open to some interpretation. So here's my take on it.
Gas pressure is the result of the movement of gas particles and the collisions they make.
To better understand why pressure exists, you need to consider the kinetic molecular theory. This idea says that individual gas particles are constantly moving and experiencing collisions. Gas particles inside of a container will collide with the inside walls of the container. Every collision between a gas particle and the inside wall of a container creates a tiny force pushing the wall in the outward direction - this is pressure.
Here is a short video which describes how to apply kinetic molecular theory to explain the behavior and properties of solids, liquids and gases.