what is circuit breaker failure protection?  what is the exact use of?

Breaker failure protection is local back-up protection. Breaker failure is initiated via protection, when the trop pulse is sent to the breaker for fault interruptin / clearing.

It is a fixed time delay relay which monitor either the breaker "a" contact or the current through the breaker to determine if the breaker has successfully interruptrd the fault current. If the BF times out, it sends a trip pulse to the adjacent protection zones and uses the other zone tripping breaker to clear the flt

Modern relays will have the ability to detect a failure in the CB and alarm/trip control centers and coordinated tripping.
BF flag can be initiated due to fixed timing (which is the least desirable), extensive arcing after trip/open, re-striking after close, single phasing and more.
The CB condition must be known for the protection to operate right when called upon

Hope it helped,