What a great song! And congrats on the wedding and learning the song. I am sure she will love it! :D

F/G is an F chord with G as the lowest note. The way it's played in the video is as follows:

1. Play the F major triad with your pinky on F (third fret of the D string,) your middle finger on A (second fret of the G string,) and first finger on C (first fret of the B string.)

2. Once this is comfortable for you, then place your third finger on G on the third fret of the E string.

These are very common chords in more advanced songwriting, as you don't hear them much in pop music. Slash chords remind me of older R&B (Michael Jackson, Luther Vandross, etc.) but you do still see them pop up. A very creative way to get from the 5 chord to the 1.

Hopefully this helps, and best of luck!

- Douglas