Play A13 one string at a time and you hear A, C#, E, G, B and F#. A is the root, C# the major third, E the fifth, G the minor seventh, B the ninth and F# the thirteenth. Said aloud, its notes are A, C sharp, E, G, B and F sharp.
From the low string to the high one you hear A, G, C# and F#. This shape drops two chord tones, E and B. They are the fifth and ninth. The lowest note is A on string 6 and the highest is F# on string 2.
A13 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 5 and 7. The second shape on this page starts at fret 2.
A13 holds the whole F#m chord inside it, A, C# and F#. Going to Dbm, one position stays the same.
A13 belongs to only one key, D major, where it is the V13.