Skeletal muscles must be arranged in antagonistic pairs so that when one muscle moves a joint the other muscle can return the joint to its original resting position. Without an antagonist muscle pairing, there would be no way for the body to return the joint to its original position.
Human muscles work by contracting. They can only pull. They cannot push. Therefore, an antagonistic pair of muscles is necessary to pull a joint in two directions.
A common example of antagonistic muscle pairings is the biceps and the triceps. These muscles are used to bend and straighten the elbow joint. When the bicep muscle contracts, it forces the elbow joint to bend, raising the lower arm. Since muscles cannot expand, the only way to return the arm to an extended, resting position is to contract the triceps. This pulls the elbow joint back to a straight position.
In antagonist pairs, muscles are often categorized as either agonist or antagonist muscles. Agonist muscles are generally associated with movement away from a resting position. In the example of biceps and triceps, the biceps are considered the agonist muscles, since they move the elbow into a bent position. Since the triceps return the elbow to its resting position, they are considered the antagonist muscles.