In Battle Islands: Commanders you level up your Rank by earning Experience (XP). 

Experience is earned two ways:

  1. Upgrading your units with Supplies
  2. Donating units in your Alliance

When you have earned enough experience you will level up to the next Rank!

Increasing your level to the next Rank will give your Bunkers and Warship more health and deal more damage.  

Even if you aren't using a unit in your Battalion, it's a good idea to continue to upgrade them for the XP bonus.