I came across and purchased a Graflex 5 Cell at an antique shop. (Unfortunately mounting it on my camera means I can never set it down properly, the flash is longer than my camera itself) I'm wondering, what do more batteries do to a flash bulb? I used my last P25 on Tuesday with my Graflite. Is the flash brighter? These cases range from 2-5 cell, what exactly does having more or less do?

Take a look at the flash bulb catalogs in the library at http://www.donsbulbs.com/cgi-bin/r/t.pl

Flash bulbs require 3 volts to fire so two 1.5 volt D cells will do the job and a 2 cell Graflite is fine if all you ever do is fire a single flashbulb.
A solenoid requires several amps to operate. If your shutter has no built in flash sync you must use a solenoid adjusted to sync with a flash bulb to trip it. 3 D cells are needed to supply enough current to operate the solenoid.

Now you want to illuminate the inside of a large conference room or similar so you need multiple flash going off at the same instant and wireless triggers are yet to be invented you set up multiple extension flash around the room and add battery extensions to your main Graflite to handle the load.
According to http://www.southbristolviews.com/pics/Graphic/manual-pdf/GrafliteManual.pdf the extension can be added in multiple to get the power your need.
You can likely unscrew the base cap and the extension tube from the main Graflite then put the cap back on the Graflite without the extension tube rendering it shorter and using fewer batteries until the need arises for the extension.