"I know whose son you are."
Translation:Ben senin kimin oğlu olduğunu biliyorum.
kim = who oğul = son kimin oğlu = whose son
senin .... olduğun = your "being". The "u" at the end is because it is in the accusative case: I know your "being".
senin kimin oğlu olduğunu = your "being" a son of someone, i.e. whose son you are.
Strictly speaking, you don't need the senin since it would still be grammatically correct to say "kimin oğlu olduğunu", however, my guess it that you probably need it in this case in order to specify that the olduğunu refers to "senin" and not the "kimin"