The Home Island is where you start the game, where you build your civilisation, and your base of operations when you do adventures.
The island is seperated into 9 Sectors, which must be taken from the Bandits that are found there. Each sector has different resource deposits that may be used to build your economy and army. Taking over the sectors also provides space and Building Licenses with which you can expand your own civilisation.
For information on how to take over your whole Home Island, this wiki has a detailed Island Walkthrough that you can follow.