"I didn't realize ___" means "I didn't notice ___" or "I didn't know ___".
You use this phrase to explain why you did something in the recent past. For example, imagine that you were in the bathroom for a long time. When you finally come out, your roommate is waiting outside of the bathroom and looks angry at you. You can say:
Oh, I didn't realize you needed to use it.
"I didn't realize" is followed by a clause, which is like a sentence inside of another sentence. Sometimes the clause can begin with "that":
I didn't realize that she had gotten laid off. If I had, i wouldn't have made those comments.
But it's often left off in casual speech:
Hey! I didn't realize you were in town!