There are two ways to invite additional members to your family. The most direct is to tap the gray "add a family member" hexagon along the bottom of the map screen. You can also tap the "Add Member" button on the bottom of the Family Roster screen, which you can access by tapping the Family icon on the upper right of the map screen.
Note that you must be an Owner to invite someone to a Family. You can see if you're an owner by finding your name on the Family Roster screen. If it says "Family Owner" next to your name, you're an Owner. This means you can invite others to the Family, promote Family members to Owners, and Rename the Family.