For fly fishing, there are two types of flies, wet and dry.
The wet flies are designed to sink and bitten by the fish underwater, while the dry lures are designed to sit on top and the fish will come up and bite them. If you are really lucky the fish will leap out of the water to do so.
The short version of how they work is to you cast them out and then gently reel them back in on still water (ponds and such) or cast them upstream on flowing water and the reel as they come back down the stream to you.
You don't need to add weight, just switch to the other type of fly if you would rather fish below the surface.