It does two things:
- Displays your Google+ profile image by your pages in search results, which increases visibility and trust in your page.
- It's also said that pages with a linked author will rank better than pages without.
Google give two ways to achieve this. One is to a link your page to your Google profile like this:
(obviously using the id number of your profile)
Note that you also need to update your G+ profile to include websites that you author - edit your profile and add your websites to "Contributor to".
Implementing this in my regular websites was a simple copy and paste.
Then I turned to my Blogger blogs. I found that Google already adds the important 'rel=author' code if your name is shown at the bottom of your posts. But it was linking to my Blogger profile (out of date and has a distant photo). The answer was simple.
If you go to the Blogger home page and click the little cog near the top-right, there's an option called 'Google+'. Clicking this allowed me to very easily switch from my Blogger profile to my G+ profile. This simple action meant that all of my posts on all of my blogs now contain the rel=author link to my G+ profile.
Google futher helps you by asking you whether you want to add your blogs to the 'contributor to' list on your profile (if you've not already added all of them in the step above).