"I love her, and I feel fine."
Translation:Jeg elsker henne, og jeg føler meg bra.
Føles is the passive voice of the verb, grammatically speaking, whereas føler is in the active voice, and so it needs to take some object. In english, the reflexive sense of the verb is implied, and so we wouldn't say "I feel myself fine," because "I feel fine" implies that it is "myself" which is fine. In Norwegian, this isn't so. Føles literally translates to "am felt," so the meg/deg/etc. becomes the subject in the form of jeg/du/etc.. So "Jeg føles bra" would literally mean "I am felt fine" which conveys the same sort of reflexive idea as "I feel myself fine." So yes, they do effectively mean the same thing, at least in this instance, but I am not entirely sure when one is idiomatically preferred over the other.