Keys in the same "Camelot key code" (like 1A and 1B, 5A and 5B, etc.) are just what music theorists normally call relative keys, meaning the two keys share the same key signatures. For example 5B (E-flat major) has 3 flats, which is the same as C minor in 5A.
The "different but harmonically compatible keys" are what we call closely related keys. Closely related keys add or substract one accidental from the main key. For instance, the keys closely related to E-flat major (which has 3 flats; again, this is 5B in the wheel) are B-flat major (2 flats; 6B) and A-flat major (4 flats; 4B).
For what it's worth, this whole "Camelot Wheel" thing is just a needless reinvention of the Circle of Fifths. (It looks to me like this Mark Davis dude just wanted a reason to slap his name on something.)