Capturing towns in Overthrow is achieved by dropping the Stability to 0%. You can check the stability in a town by using the Y menu, clicking on "Map Info" top left and then clicking on a town.
If the town has a population under 100, NATO will abandon the town when stability is zero. When above 100 it will trigger a QRF, if resistance wins the battle then NATO will abandon it. All online players will receive Influence when a town is abandoned, and towns controlled by the resistance will generate taxes.