I get it when I see both the man and women showing their "modesty" by him wearing his whites, head covered, and she is also that makes sense.

What I do not understand is when he is in shorts, tank top, ie dressing not modest at all and his wife and kids are all in black.

Can someone give me the exact quote as some people I have heard say it is not in the Koran that women have to cover up?

Interestingly in the bible you are supposed to cover your head.