The Entertainment District increases your empire's happiness by providing a place for your citizens to unwind through activities such as casinos, restaurants, clubs and other amusements. The higher the building level the more happiness is produced. Be careful, it can also produce a lot of waste, which could negate some of the happiness it produces.
The Entertainment District has a Lottery that allows you a chance to win some Essentia by clicking on links to game rating sites. Then when the page comes up, vote or rate the Lacuna Expanse. Each time you click on a link you'll be issued a lottery ticket (virtual, you don't actually get a ticket) giving a chance to win that day's lottery for the zone. You can click on each link again for another chance 24 hours later.
Every day, each zone has a winner who gets 10 essentia! If you happen to have 5 entertainment districts, and they are all in different zones, you could then vote on one or more links per zone, thus giving yourself 5 zones to win in. Or, you could vote on all the links in one zone, giving you maximum chances to win in that one zone.