What you face here is unusual word order together with dative instead of genitive:
Einem reichen Mann, dem wurde seine Frau krank. (speech, a lot of focus on the rich man)
Einem reichen Mann wurde seine Frau krank. (speech, focus on the rich man)
Eines reichen Mannes Frau wurde krank. (genitive, focus on the rich man)
Die Frau eines reichen Mannes wurde krank. (standard)
Dative is normally used instead of genitive in speech only, so there aren't too many rules about this. Dialects rule it, and fairy tales are captured as spoken, with only the most unintelligible dialect phrases removed.
