The physics...

The ground glass (or plastic) screen acts as a diffuser, scattering light passing through it randomly rather than simply letting it pass through unaffected. An image can be brought to focus on the screen by adjusting the lens, and the image we perceive comes from scattered light that is traveling in the precise direction of our retinas.

The scattering quality is what makes it work. It's a fundamental of how we see things. If light (photons) falling onto an object bounced off it in an ordered fashion, in a certain direction, you could only see that object if you were looking at it from the same direction the reflected light was traveling in. This is how mirrors work. On the other hand, photons scattering randomly off an object with a non-reflecting or non-transparent surface (generally most things we come into contact with) allow us to see that object from many different angles.

It's a very similar process to seeing a film at the cinema. Light is projected onto a diffuse surface which then scatters in any direction it can, and we see those photons that come precisely our way. A back-projection system is a good analogue of what we're seeing with a ground-glass screen, the screen has to diffuse light in order to form an image, but also allow the diffused light to pass through to the observer. We then perceive those scattered photons coming our way the same way we'd perceive any regular "real" object. If you tried projecting onto a completely non-diffuse surface an image won't form as there is no scattering of light that will then randomly come our way, to be intercepted by our retinas. The only completely non-diffuse surfaces are logically either reflective or transparent.

Back to camera things, the focusing screen sits at exactly the same distance from the lens as the image sensor or film-plane, so when light is brought to focus on the screen it will also be focused on the sensor (or film). Without the ground glass layer we wouldn't have this reference point during manual focusing - and while our eyes may be able to bring a scene into focus, in the absence of the diffusing layer there would be no guarantee that the lens was also correctly focusing the intended image on the sensor (or film).

This may not be the clearest of explanations - any edits welcome if it makes it more coherent - but hopefully you'll get the idea...