Play C13 one string at a time and you hear C, E, G, Bb, D and A. C is the root, E the major third, G the fifth, Bb the minor seventh, D the ninth and A the thirteenth. Said aloud, its notes are C, E, G, B flat, D and A.
From the low string to the high one you hear C, Bb, E and A. This shape drops two chord tones, G and D. They are the fifth and ninth. The lowest note is C on string 6 and the highest is A on string 2.
C13 is a closed shape with no open strings. Move it to another fret and the chord changes, but the fingering stays. Strings 5 and 1 are not played. Everything fits between frets 8 and 10. The second shape on this page starts at fret 3.
C13 holds the whole Am chord inside it, C, E and A.
C13 belongs to only one key, F major, where it is the V13.