2 uses are common. One, they mean that they were on call last night, are tired as hell and are just in the hospital to round and finish their $hit. "God, I'm post-call and I can't wait to get the F outta here!"
Second, and more applicable to students than residents, they mean they get the day off after being in the hospital overnight. "Yes! I'm post-call, who else wants to go to the amusement park"