leader or backing?
When you asked about the "line attached to the floating line," were you talking about the leader (the monofilament, usually tapered, connection between the line and the fly) or were you talking about the backing line (the braided line behind the floating line)? Others have answered if you meant the leader. However, if you meant the backing line: Backing line is used so that if a fish is strong enough to pull out more than the fly line, there is additional line to allow the fish to run. For trout and similar size freshwater fish, it's rare for the backing line to be needed. I've had only a few trout run that far in 45 years. Normally, 50-100 yards of backing is used with trout size fly tackle; 150-200 yards for salmon and steelhead. Reels should be chosen to hold that much backing. A lesser advantage is that it prevents the back end of the fly line from being wound around a small reel hub, from which the fly line would come off looking like a coiled telephone cord.