"What is beat leasing?"

Leasing or licensing beats can be viewed as renting music from a producer. 

Beat leasing allows you to incorporate a producer’s instrumental/beat into your song while the producer maintains full ownership of the copyright in the beat.

"What do the terms ‘exclusive’ and ‘non-exclusive’ mean?"

When you lease a beat, the producer will grant you either an exclusive or non-exclusive license to use the beat in a new recording.