According to a recent study, not much. Twenty-nine patients with new ankle sprains were in the study. The authors found that a measure of ankle swelling wasn't related to patients' ability to do sports and daily activities. Swelling also wasn't related to whether or not patients could put weight on their hurt ankles.
The authors believe that patients' own reports are the best gauge of ankle function. This makes sense given the number of personal factors that go into how patients recover from injuries.