Cycling is an activated ability you can use only while the card is in your hand. By paying the cycling cost (listed on the card) and discarding it, you draw a card. It turns otherwise dead cards—situational removal, expensive bombs, lands you don't need—into fresh draws, smoothing your draws throughout the game.
The most common misunderstanding: cycling is not casting the spell. The card goes to your graveyard via discard, so it never enters the battlefield and "when you cast" or "enters" triggers don't fire. However, cycling itself is an ability that uses the stack and can be responded to, and many cards have "when you cycle this" triggers (like Astral Drift or Zenith Flare payoffs).
In Commander, cycling adds consistency and graveyard fuel, enables typecycling tutors (Decree of Pain), and pairs nicely with discard-matters and graveyard strategies.