The cloaking device need not be remotely functional, only present. The ship can be heavily damaged, and still successfully perform a cloak intercept. Even Ion storms and Nebulae do not stop a cloak intercept.
After the ship has successfully intercepted its target and one or both ships are hostile to each other (be it Primary Enemy or Kill mission), it will first engage its target fighting from the right hand side and then (if it survives) fight in the regular battle order.
The most common use of this mechanism is to remove ships from enemy fleets, especially when they're in deep space, sacrificing your ship to destroy the enemy's ship. For instance, destroying a Cobol or NRS in an enemy fleet can cripple an entire offensive, buying you critical time to mount a counterattack, or by forcing the enemy to send fewer ships.