You access car mode by dragging down from the top of the screen with two fingers to display all the option buttons... The car mode button should be somewhere in the group of buttons that you see.
The main idea of car mode is to limit what you can do on the phone, and to make what you can do easier. You get a screen with four large buttons. (One for each feature that you're allowed to use). They are easy to see and press. Samsung threw in an extra "find your car" feature that uses GPS if you need help in remembering where you left your vehicle.
I don't touch the phone at all when I'm driving. The phone sits in a pocket on my center console and I use Bluetooth for incoming/outgoing phone calls, so car mode is of no use to me. If you want to text, and use Facebook, Twitter and access other social media and apps while you're driving, then car mode isn't for you.
Below is what you see on the screen when car mode is activated: