Antagonist training just means alternating exercises that target opposing muscle groups, like chest and back, or biceps and triceps, etc. The list of benefits includes: quicker recovery, greater strength levels and shorter workout times.
Antagonist training allows you to recover more quickly between sets due to the arrangement of the nervous system. When you maximally activate a muscle group, the nervous system inhibits the opposing muscle group for greater movement efficiency. This phenomenon decreases the time necessary for recovery and it helps restore strength.
Example: Arm Training
When you do a set of biceps curls, the triceps are forced to relax so they don't oppose the elbow flexing action of the biceps. This is accomplished by a "loop" within the nervous system structure – when certain motor units are activated, others are inhibited. If this action didn't occur, you wouldn't be able to move at your joints because each set of opposing muscles would be attempting to contract against each other.
This design can be used to your advantage. If you alternate exercises for opposing muscle groups, the nervous system will inhibit the muscles that aren't being worked and you'll recover your strength more quickly. So if you're working your chest, alternate it with a rowing exercise. Simple. You can even pair overhead presses with pull-ups or lat pulldowns to get the same effect.
Think of antagonist training as a mini yoga session for the opposing muscle group. You'll be able to perform your chest/back cycle in less time than if you performed each body part separately, and you'll be stronger.