The notes of E13 are E, G#, B, D, F# and C#. E is the root, G# the major third, B the fifth, D the minor seventh, F# the ninth and C# the thirteenth. Said aloud, its notes are E, G sharp, B, D, F sharp and C sharp.
From the low string to the high one you hear E, B, D, G#, C# and E. The root, E, shows up twice in the main shape. F#, the ninth, does not sound in this fingering. The lowest and highest notes are both E, two octaves apart, on strings 6 and 1.
E13 lets three strings ring open. String 6 gives E, string 4 gives D and string 1 gives E. Everything fits between frets 1 and 2. The second shape on this page starts at fret 12.
E7 is one change away. String 2 is left open, bringing in B, the fifth of E7. E13 holds the whole C#m chord inside it, E, G# and C#. The change that keeps the most fingers still is to E. Two stay in place.
E13 belongs to only one key, A major, where it is the V13.