You have to consider how stereo works, and the two different ways that it can be represented. Normally it looks like two very similar, but different channels. The position of anything in the stereo field is represented by its relative level in each channel. So if you had something that was completely on the left or right, it wouldn't turn up at all in the opposite channel.
But that's not the only way you can represent it. You can also look at it as two separate channels, one of which is the center information - which is what you get if you add both channels together (the Sum) - and the other of which is what you are left with if you subtract one from the other (the Difference). Yes, as a result of this there will also be phase differences between the two signals. When this technique was first invented by Alan Blumlein, the names for the two channels were slightly different - the Sum channel was called the Mid, and the Difference channel was called the Side. One thing it's important to note is that you can do the transform from one to the other without loss. So why would you want to do this?
Well, if you convert from stereo to sum and difference channels and do nothing, the result will clearly end up being the same when you convert back. But if you convert to sum and difference, and reduce the level of the difference channel and convert back, you'll end up with more mono signal and less of what makes it stereo - in other words, the stereo width is reduced. And if you raise the level of the difference channel, then the stereo will be wider than it was.
So what the stereo expander does is - Convert from stereo to sum and difference, and give you a slider which lets you alter the level of the difference channel, and then convert back to two-channel stereo again. But there's a sneaky bit; the center channel pan fader. This takes the mono sum signal and, when it converts back to stereo, allows you to reposition that mono signal so that it isn't central any more. Personally I've never found a use for that but hey, it's there if you really need it. That bit is slightly clever though, because it has to allow the actual stereo signal to be recreated more or less as it was.
Anyway, that's the basis on which it works. There are other things you can do with it as well, but that all gets slightly more complicated. Also it's a technique commonly used for stereo microphone recording - you can use a ribbon mic to pick up just the side information, and either an omni or cardioid (depending upon what you want from the signals) for the sum. And then you can decode them back to stereo, with width control.
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