At least one undrafted free agent usually makes the roster on each team in the NFL every year—sometimes two. Nate Eachus that made the roster last season as an undrafted free agent.
There is a flurry of signings after the draft, and there is a lot of good talent looking for an opportunity. Sometimes all it takes is giving a player a chance, time and good coaching.
Although rare, good players sometimes slip between the cracks. Tony Romo is probably the most well-known undrafted player currently playing in the NFL, but there are dozens of other examples.
John Dorsey and Andy Reid can only make over the roster so much at the top because of the contracts, but the last few guys that make the roster can be guys that fit what they want to do. That's where the draft and these undrafted free agents can make a difference.