Play C6/9 one string at a time and you hear C, E, G, A and D. C is the root, E the major third, G the fifth, A the sixth and D the ninth.
From the low string to the high one you hear C, E, A, D and G. The lowest note is C on string 5 and the highest is G on string 1.
C6/9 is a closed shape with no open strings. Move it to another fret and the chord changes, but the fingering stays. String 6 stays silent. Everything fits between frets 2 and 3. The second shape on this page starts at fret 7.
C9 is one change away. String 3 moves from fret 2 to fret 3, bringing in Bb, the minor seventh of C9. C6/9 holds the whole Am chord inside it, C, E and A. Going to F, two positions stay the same.
You will find C6/9 in six different keys. In C major it is the I6/9, in F major the V6/9 and in G major the IV6/9. The rest appear in the key list.