The notes of F13 are F, A, C, Eb, G and D. F is the root, A the major third, C the fifth, Eb the minor seventh, G the ninth and D the thirteenth. Said aloud, its notes are F, A, C, E flat, G and D.
String by string, low to high, it plays F, Eb, A and D. This shape drops two chord tones, C and G. They are the fifth and ninth. It runs from F on string 6 up to D on string 2.
F13 uses no barre and no open strings, so you can slide the shape up the neck without changing fingers. Strings 5 and 1 stay silent. Your hand covers three frets, from 1 to 3. There is one more position, starting at fret 8.
Fm7b5 is very close. You only move strings 3 and 2. Inside F13 sits a complete Dm chord, made of F, A and D. F shares two positions with this shape.
F13 belongs to only one key, Bb major, where it is the V13.