The Map lets you find and locate Alien Bases. You have to eliminate all Alien Bases to stop the alien invasion.
When you attack Alien Bases you earn stars and resources. The number of stars you earn is based on how complete is your victory over the aliens. You win the battle by earning at least one star, which let's you advance to the next Alien Base. After all Alien Bases in a Map Area are defeated, and you have earned enough stars, the next Map Area is unlocked.
To attack an Alien Base you need to use Fuel, which is replenished every 5 minutes, and can also be obtained through Daily Rewards as well as gifts. Defeating Alien Bases also awards you with Shards, which you can use to buy and upgrade special buildings for your farm.