How do zener diodes regulate voltage?
It is easiest to understand just by looking at the following curve:


The vertical axis is the current, you can see that in reverse bias "breakdown" region, that no matter how much current you flow (unless you are flowing a small amount) that the reverse voltage changes very little (see the breakdown voltage label on the graph).

That means that this circuit has very low compliance, i.e it doesn't move (in voltage terms) very much with changes in current. This is the characteristic of a nice regulated voltage supply.

Now obviously there are limits to the amount of current and the amount of regulation, but as a first order system it works well.