While I have no specific knowledge of Falcon 9's fairing, it is common for compartments of space vehicles to be purged or pressurized slightly above ambient while on the pad to keep contamination out of the vehicle.
After launch, vents allow the pressure in the compartments to drop in a controlled fashion as ambient pressure drops.
To get a feel for how this operates, you can read about the shuttle's comparable systems here. Some excerpts:
Three purge circuits are connected by the T-0 umbilical to ground
equipment before launch during the preflight countdown and postlanding
phases. Purge gas (cool, dry air and gaseous nitrogen) is provided to
three sets of distribution plumbing: the forward fuselage, orbital
maneuvering system/reaction control system pods, wings and vertical
stabilizer; the midfuselage; and the aft fuselage. The purge gas makes
all the unpressurized volumes inert, maintains constant humidity and
temperature, forces out any hazardous gases and ensures that external
contaminants cannot enter.
At T plus 10 seconds, all vent doors are commanded open. At T plus 80
seconds, vent doors 8 and 9 are commanded closed to prevent hazardous
gases from entering the aft fuselage; at T plus 122 seconds, vents 8
and 9 are commanded open. All vent doors remain open during the
remainder of ascent and on-orbit operations.