There isn't enough research out there to explain the role of autapses however having read a selection of the latest research I can perhaps explain some of the proposed theory. Autapses may self inhibit or self excite. In the latter, one of their roles is thought to be to make a rhythmic action potential. This allows the brain to have an action potential which fires rhythmically if needed. Autapses of this nature have been identified in Interneurones for this purpose of continual activity.
Then there is even less researched areas. Imagine you need a neurone to fire twice to certain neurones but once to others or other similar requirements. A mixture of autapses with self inhibition and self stimulation properties can achieve this elegantly in a short space.
Things like this however aren't required frequently. Usually interconnecting neurones is sufficient, however possibly in places of high rhythmic activity or fine control these autapses may be helpful.