Why would an F16 battery make a buzzing sound?
The relevant part from the video description that you referred to:


  You'll here a buzz, thats the battery, then you'll here it spooling thru N (nancy) stages.


The word "spooling" does not refer to the battery, it refers to the engine. The battery is static and does not spool.

There are several measurements or indications used to convey a turbine engine's state. One of them is N2 (others include N1, Engine Pressure Ration, Exhaust Gas Temperature, Interstage Turbine Temperature and several others).

Back to the video description. In this case I believe the author was referring to the N2 measurement that raises ("spools") and needs to be monitored during engine start.

N1 is the low-pressure compressor speed, and N2 is the engine's high-pressure compressor speed (both usually expressed in percentages rather than actual RPM).

More information about N1 and N2 can be found here: What are N1 and N2?