In order to invite people into a sphere, you must be the administrator of that sphere.
Click on the "Invite someone" button in the main header.
or from within the sphere, on the "Invite members" link in the right hand column.
In the popup that opens, fill in the fields, then send the invitations (by clicking on the "Invite" button).
You can invite your contacts by their Whaller username (if you know it), or their email address.