"František's aunt likes my hands."
Translation:Františkově tetě se líbí moje ruce.
If your question is "why not genitive (Františka)?": "Františkově" is a possessive adjective, something that is used only for proper names (a bit like Cesarean for Caesar or Neronian for Nero – Franciscean?). Its nominative singular is "Františkův"; it is in dative here to agree with "tetě."
The grammar of líbit se, with dative, is explained excellently by Endless Sleeper's comment.