If you're rested, you're wide awake after a good night's sleep. Rested students tend to score better on tests than those who stayed up too late the night before.
If you were choosing the pilot to fly your airplane or the surgeon in charge of a relative's operation, you'd pick rested people, who are refreshed by plenty of sleep and are wide awake and prepared to do their jobs well. Someone who's rested is alert and invigorated, as opposed to a tired or fatigued person. The Old English root is ræste, "sleep, bed, or mental peace."