When you run an ad on Facebook, the default channel is Facebook itself: users browsing through the newsfeed on their laptops, smartphones, or tablets. You are reaching people either using the Facebook app or on Facebook’s website.
The Facebook product team had the brilliant idea to expand the reach to people using other mobile apps (like games) and browsing other websites. So the Audience Network was created.
As an advertiser, the main advantage is that you can leverage Facebook’s targeting capabilities. If you want to promote a new 3 in a row game, you can target people playing Candy Crush (assuming that Candy Crush has the Audience Network SDK installed.)
For publishers, it is a new way to monetize your content or mobile apps. Since Facebook keeps track of each user interests, ads on your website or app should be more relevant (if the advertiser did a good job matching interests and ad creatives.)