Pretty simple. The liquid burns, but there's no pressure, so it needs to vaporize. Same concept as a canister stove, or alcohol stove for that matter.
You don't have streams of liquid flowing from your canister burner nozzles, for example, it's the vaporized gas that comes out that is fueling the burner when operating properly. A canister is, after all, filled with liquid when under pressure.
In a liquid stove, in order for the liquid fuel to vaporize, it needs to be heated up to the point of boiling off and turning to vapor. This happens in the "generator", typically a short piece of piping that runs directly over the burner. When the burner is going, the fuel flows through the generator and gets vaporized by the heat of the burner itself, and the gas given off is what burns, not the liquid.
In order to start this process though, liquid stoves need to be "primed". An MSR has a little pan under the burner, a small cup, some have a wick in there too. You start by pressurizing the tank, then crack the valve open just enough for a little liquid to flow into the "primer pan", then shut the valve off. Light the primer pan with this small amount of controlled fuel in it, and when it gets the generator up to temp, slowly crack the valve open again. Typically there will be some sputtering at first, as a little liquid seeps in, and eventually it will take off. Then the process is self sustaining.
If it's not hot enough when you crack open the valve again, the fuel won't vaporize and will just spill out, and over, the cup, possibly creating quite an impressive ball of flame. We've all done it once, at least.
It's kind of an art, as well as a science, so practice a little… in a safe environment obviously ;)