How do I take tiles in other States, after teleporting there?

Tiles can only be taken next to your own / alliance member tiles or one of your alliances structures.

The only way to get started in another state is during the alliance wars. Once you have a war, place a rally point structure in the enemy state. Use this to start taking tiles. You will need to do this to attack the enemy alliance centers and after the war you can maintain tiles in the other state and travel back there at any time to take more.