Google Authorship is a way of enhancing the way your writing appears in Google’s search results. If you’ve told Google where to find your writing by connecting it to your Google+ account, your posts and pages will show up with a byline and your Google+ profile picture, like so:
Why would you want this? It makes your posts more visible, and as people see your posts more often, they will be more likely to associate you with your work. It’s like a super-charged byline. Plus, I’m sure it has some SEO value or something.