As jayk stated the heat from the flame causes the cells in the plant to burst, effectively killing the plant, but it does nothing to the plant roots and many of these "weeds" will grow back from those roots.
If you use one of these "flame weeders" due care in use must be observed since imporper use could, and has, cause a large fire.

Yeah, I don't think I'm going to get one. Sounds too dangerous. I'm just looking in to all the different weeding tools out there.

"Weed" burners, propane torch flamers are not an environmentally sound way to kill "weeds" since they use a non renewable resource to do that.
A far better, more ecologically sound, method of unwanted plant growth control is using mulches of materials from renewable sources.

The propane weed burner was first used to kill weeds in cracks on hardscapes.
Then the companies found a market with people who did not need it for parking lots & side walks.
It is better the round up or gasoline, but there are so many other ways to kill weeds.

An article in the current issue of Garden Gate magazine talks about starting a fire while using one of these weed burners.

Like anything else, this product needs to be used with some common sense. Best use is for weeds coming up in gravel and driveway cracks, etc. It's also good for annual weeds coming up in turfgrass. I would not recommend it for use in dried bark or wood chip mulch and certainly not in pine straw.

Some people will stick their fingers in the garbage disposal.