A person cannot jump higher on the moon than on Earth because the Moon has only about 1/6th of the gravitational force that Earth has. This means that there is less force pulling the person back down to the ground on the Moon, but it also means that there is less force pushing the person up when they jump. In other words, the reduced gravity on the Moon both decreases the force of gravity pulling the person down and decreases the force of the jump pushing the person up, resulting in a similar jumping height as on Earth.
