Ah the old slash chord. Lots of theoretical fun to be had. As others have mentioned, concretely, the formula is: Chord/Bass Note. So when in doubt, play the note after the slash and all will go well.
WHY they use slash chords is a bit deeper.
Reason 1 is pretty simple: to play a chord inversion, usually to make the bassline move more smoothly. Take a C/E slash chord. E is the 3rd of the C chord, so all that means is it's a C chord with the 3rd in the bass as opposed to the usual Root in the bass. If everybody else in the band is playing notes in a C chord, and you the bass player are playing a lower E than anyone else, mission accomplished, you're all playing a first inversion C chord. This is usually used to make a bassline walk smoothly through changes. A classic country and rock example which you have probably already played hundreds of times would be G - D/F# - Em. The cord progression is G - D - Eminor, while the bass politely walks down from G to F# to E. in your example, the Fadd9/A is actually the simpler of the 2 slash chords, since it's just putting a different chord tone in the bass than the expected F. If you told us what chords came before and after, we could probably deduce why he chose to use that inversion, I'll bet it makes the bassline flow smoother, nothing more than that.
The other use of slash chords is less simple. They're used to create convenient voicings and names of more complex chords. G/A is an excellent example of this. Concretely, that means a G major triad, with an A in the bass. But wait... the note A has nothing to do with a G triad. You might think, well, they're just going for a Gadd9 sound, since A would be the ninth of a G chord. Nope. Putting the A in the bass does not give you that sound, and that's not the intent here. You can't think from G in this case, you have to think from A. G/A is actually a simple way of writing A7sus2sus4(omit5). Or A11(omit 3 omit 5). Or any other messy way of writing the same thing. G/A is a very nice suspended dominant chord voicing, great to use if you're going for a gospel/church music sound, and leads very nicely to D major.
So to sum this up, if the bass note after the slash is a chord tone, mystery solved, you think from the chord since you're just playing an inversion of that chord. If the chord and the bass note are not related, then you think from the bass note and try to figure out how the chord relates to it. I've googled around a little, and surprisingly, haven't found a single, simple, correct resource on slash chords and their equivalents. Somebody tried on the gear page, but I see at least one error in that list, so I can't recommend it. Anybody else know of one?