My father made me who I am today.

Literal

Father [subject-が] me [object-を] present's me [into-に] did-for-me.

Xを Yにしてくれた = 'made X into Y as a favor.' 現在の私 ('the present me') is slightly more formal than 今日の私 ('today's me'). The のだ at the end adds explanatory emphasis.