The law at issue reads:
Sec. 3.1‑10‑5. Qualifications; elective office. (a) A person is not eligible for an elective municipal office unless that person is a qualified elector of the municipality and has resided in the municipality at least one year next preceding the election or appointment;
The appellate majority held that the second prong was not met, and that Rahm does not qualify to run for office since, in their interpretation, he hasn't "resided" in Chicago even though he maintained his legal residence there and remains qualified to vote there (which, likewise, is based on a concept of residency).
I don't have anywhere near the experience of people like Rick Hasen when it comes to this little corner of the law, but campaign and elections law is one of my areas of focus (I'm currently the president--ironically appointed by executive fiat--of the Georgetown Election Law Society). And, for whatever it's worth I'm fairly sure that Rahm gets to run. I think the appellate majority misinterpreted the statute at issue. My reading of the statute (and I'm willing to bet the reading that the Ill.Sup.Ct. will adopt in the next day or two) is that being a legal resident of Chicago for the year prior to the election--which Rahm was--is all that's really required.
If you want a better discussion than this of the case, check out Rick Hasen's article in Slate. Richard Winger has Rahm's brief here.