Trailer classes are a quick way of knowing how much weight your vehicle can handle safely. Class I is the lightest weight, class II is a little bit more.
Googling trailer classes led me here:http://www.autoanything.com/towing/how- ... class.aspx
While the manual does mention towing, keep in mind that it covers all the A3's, not just the E-tron. At least that's the case for the on-line manual.
I tried to get my dealer to add towing to my car before delivery. They wouldn't do it, even after I showed them the European build pages where it is a option. My guess is that in the European market, people tow a reasonable load - say small trailers or put a couple of bikes on the back. In the US market, folks see a receiver for a hitch, and they'll hook anything up that fits. I know I have.
Total passenger and cargo capacity is maxed out at 450 kg/992 lbs. per the loading sticker on the b pillar. That doesn't leave much towing capacity once you put people and a bag or two in.
Our UK friends are discussing this in more detail here - https://speakev.com/threads/tow-bars.11231/