What is "the tab"? Whose tab are you looking at? There is no banjo on the recording, so anything goes.
If you want to play in the key of F, one way is to do as you've described: capo at the 3rd fret and tune the 5th string to C. Many people do prefer to play in the key of F without any retuning or using a capo at all. That sounds very nice, too.