When the sniper raises himself up to shoot the man in the armored car and the old woman informant who points out his location, he makes himself a target of the enemy sniper across the street. His enemy takes the opportunity and fires a round. The Republican sniper is hit in the "right forearm," and this injury prevents him from being able to hold his rifle. He literally cannot lift it with his injured arm.
A rifle is a long gun, usually with a sight on it to help the user take aim, and it requires the use of two working hands. Because the sniper's right hand just won't work anymore, he cannot use the rifle. This is why he must switch to using the much smaller revolver, which can be held and operated with one hand.
He identifies the place where the enemy sniper's bullet entered his arm, but there is no hole on the other side of his arm where the bullet left his body. This allows him to realize that the bullet has likely become lodged in his bone, fracturing it. This explains why he has lost use of the arm. In fact, at first, he does not even feel pain in the injured appendage, just a "deadened sensation," as though the arm were no longer even there.