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To use ground glass, I turn the focusing ring on my lens until the image becomes clear on the glass. But what is it about the glass that makes this work? What are the physics involved?

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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...

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The ground glass, or focusing screen, is, in the simplest case, literally just a piece of glass that has been ground, so one of its sides has a rough/matte surface (see also the article Ground glass on Wikipedia. It's actually quite easy (at least for large format cameras) to make ground glass yourself, see this how-to for example.

The ground glass just acts as a projection screen, this is due to its translucency*, i.e., this enables you to see what the lens projects onto the ground glass (directly in large format cameras, or with some additional mirrors in SLRs and TLRs). The actual focussing happens by moving the lens's focal plane into the right position with respect to the ground glass, see for example this answer.

There are more sophisticated types of focusing screens that involve split screens or microprisms. For an explanation, see this article.

*) Note that translucency is the "scattering quality" as nicely explained in Darkhausen's answer to this question.

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