Go to the Eagle's Nest and choose a Settlement you want to capture, then tap the red "Raid" button below the description of the Settlement. Make sure you spy on the Settlement before the attack! You can do that by tapping on the eagle icon, situated below the description of the Settlement.
Once you capture the Settlement, send some reinforcements to strengthen your position there. This can be done as follows: go to your Keep, open the "Settlements" tab, select the captured Settlement, and tap the arrow button on the right. After that, choose the defensive Units you would like to send, and tap on the red "Send" button.