Being in an Alliance is a pre-requisite to use the reinforcement feature in the game play.
Once request for reinforcement is sent, it will appear in your Alliance chat. Your Alliance members will be able to see the request and donate troops to your cause.
The number of troops you can receive will change depending on the level of your Alliance.
Once you've received your reinforcements, add them to your army by tapping on the new "Reinforcements" option in the load-out screen before you begin your battle. Each Alliance can hold a different number of reinforcements based on their current level.
During the battle, summon the reinforcements by tapping the 'Summon' button in the bottom right corner after you begin your attack. The reinforcements will deploy automatically, and will have a "+" symbol above them on the battlefield.
Reinforcements are temporary units, who have been sent to help you for just one battle. After they've fought, they'll leave your army, regardless of whether they survived or not.