Combined reference to multiple leave
Leave may be used as a countable or uncountable noun when referring to the period of time one is away from a job.
Leave of absence is granted. (uncountable)
Leaves of absence are granted. (countable)
To answer your supplementary question, leave in "leave of absence" means permission. It also occurs in the phrase "by your leave": "with your permission".
Leave has come to mean absence in its own right, and to use it to mean permission as in the examples in ODO is archaic.