The letters above mean the implied/intended harmony. There are 2 possibilities I see here as to why an F and C7 are listed.
If you are playing solo
There are certain common chord progressions in music. The go-to example always seems to be I-V-I, and what this means is that the chords move from the first/tonic chord of the scale(in this case F, to the fifth/dominant chord of the scale (in this case C or C7) then back to The First.
You'll notice that there's no C present in the C7 either, but it does have other notes of the C7 chord. This is an example of an implied harmony, because the function of the chord is to be a dominant so it can lead back to the root in the next chord(the F in the bass).
It's much more common to think of a chord progression as V - I - V (as it is in the first bar here) than it is to say V-iii-ii(as it would be if we take the bass to be the root notes) so what the transcriber has done is to write the V-I-V chords instead of the V-iii-ii chords(which would be C Am Gm). As you can see, C Am G has a different harmony and feel to C F C7, but C F C7(or V I V) is the intended harmony underneath.
If you are playing as part of a group
the chords listed here are what the overall chord progression will be when everybody is playing. A C consists of CEG. An F major consists of FAC, and C7 is CEGBb. While you are playing the A, another part will perhaps be playing F or C, or both!
Thanks for that. When I first looked I thought it seemed simple, but there are a number of added little complexities in there that can throw you, for example you're not beginning on the root. If anything is unclear let me know in the comments and I'll update :)