Hi Elizabeth!
It basically provides in-depth statistics, like an extended version of Etsy's shop stats. I use it for my sites, and I think it's pretty useful.
Go to analytics.google.com and work for there. Once you've set up your site's profile, you should see a page with code and a funny-looking number. Then go back to Etsy, then Shop Settings > Options > Web Analytics. Paste the number/letters into the Web Property ID field, and you're set to go!
You'll have to check your Google Analytics on Google by the way, not Etsy.