The Atomic transporter is for "jumping" to different islands. The transporter has a limited range and is NOT attached with the other one on the same island. Where the islands are close together you are able to see the other island but you can not attack.
You do not need to control the Transporter to get the bonuses.
If you control all transports of the same color then you get the bonuses of that color.
Be careful eliminating a player. Don't count on using the elimination bonus right away you must wait until your next turn to use the extra troops!