A silo in its unlaunched and launched state
To launch a missile from a silo, put an infantry or mech on it. Afterwards select the infantry again and click on launch. Then you are supposed to click on a tile. The missile affects an area, which concerns the tile you clicked on and all tiles bordering to that tile and all tiles bordering to those tiles. In totaly 13 tiles will be affected (if you did not launch it near the edges of the map). All units in that area will get 3 hp damage.
Note that if you click on launch, you will not see on the next screen that you are about to launch a missile. So if you decide you do not want to luanch the missile after you clicked on launch, a way to cancel is to return to the yourturn page and reload the game.
After an infantry launched a missile, the infantry can not fire on nearby units. Also firing first, unables your infantry to launch the missile.