The chemistry is a bit over my head, but based on various MSDSs for brake cleaners, the leading chemicals are acetone and some type of naphtha variant. Naphtha is a highly distilled petroleum product that evaporates relatively quickly and doesn't usually leave a residue (looks to be dependent on quality).
As far as why one rusts and the other doesn't, there could be a multitude of other reasons. There are better tests to determine if brake cleaner leaves a residue. First thing that pops into my mind is Simple Green is concentrated, whenever I use it I dilute it with water. Water obviously promotes rust/flash rusting.
Based on what I've been reading it could also depend on the brand, some using cheaper chemicals than others that might leave residue.