As Adrian clarified you attach some monofilament to the end of the flyline on the 'business end' because that's the end the fish sees (or doesn't see in this case). This is referred to as the leader and you can get them in one piece tapered for $3-5 ea. or tie your own using gradually lighter line as you go forward. A hand-tied leader is composed of three parts: (1) the butt (2) the taper and (3) the tippet.
The tippet is the finest part that is attached to the fly. It is often replaced or changed to meet conditions whereas the butt is never changed and the taper changes number somewhere in-between.
It's typically about 20% of the overall length of the leader.
You can also use a 'bite' tippet which is made of wire for toothy fish.
Those little barbed inserts would probably work for small fish but I would not trust them while fishing for anything of size. Learn the nail knot and you'll be set.