This sentence is grammatically correct in French, but it's a bit of a riddle! It's meant to be confusing, playing on the fact that a father is also a son.
Here's how to break it down:
* My father is the subject of the sentence.
* The son of my mother and me is the object, referring to the speaker.
* The riddle is that the speaker is also a son of his/her mother.
Essentially, the sentence is just a roundabout way of saying "I am my father's son".