Learn Turkish: Ben farkında değildim.

The root word is fark, meaning 'difference' or 'realization'. The concept of being aware in Turkish uses a fixed expression: farkında. It literally translates to something like 'in its realization'. It breaks down as fark + -ı (its) + -n- (a buffer letter) + -da (in/at).

How does the word değildim break down to mean 'I was not'?

It is built from three pieces. Değil is the basic word for 'not'. To make it past tense, we add the suffix -di (was). Finally, we add the personal ending -m for 'I'. Put together, değil-di-m translates to 'I was not'.

Do I have to keep Ben at the start of this sentence?

No, it is entirely optional. Because the ending -m on değildim already clearly tells us the subject is 'I', you can naturally just say Farkında değildim. Using Ben simply adds extra emphasis.

If I wanted to say 'You were not aware', how would the sentence change?

You only need to change the personal ending on the last word. Instead of -m for 'I', you would use -n for 'you' (informal singular). The sentence becomes Sen farkında değildin.