Original: But when a relative key ascends the opposite side of the circle from its home key, then theory suggests that double-sharps and double-flats would have to be incorporated into the notated key signature.
Simplified: But when a relative key is on the opposite side of the circle from its home key, then in theory modulating to that key would involve adding so many sharps or flats that double-sharps and double-flats would have to be written in the key signature.