I've thought this over for a long while, and this is what I can understand: Adding "up" suggests a certain result. In this case, "to rest up" suggests that he rests in order to regain his energy. The result of the "rest" is a re-energised body/mind. The verb "rest" by itself doesn't imply this result as clearly. This can be applied to other phrasal verbs ending with "up", and I've seen "out" and "off" suggest the same "certain result". However, not every phrasal verb with "up (or off/out)" follows this meaning, so be very careful! You still need to learn the phrasal verbs as you go. Phrasal verbs rely on context, and so their meanings are not automatic.