The company being not publicly traded means just that: their stock is not sold in regular markets (v.g. Wall Street).
A company that is publicly traded benefits from access to bigger opportunities of financing by selling stocks, but also must give more information to the public and authorities so the public can know what they are buying.
As I understand it, Earle's plan was to make the company go public, and emit more stock. This would have made the share of control of the original owners to dilute.
Earle's benefits from this by two ways:
- buying a significat percentage of the new stock, increasing his personal share
- more importantly, since he thought that Wayne would not understand the maneuver (after all he was just a frivolous playboy), he expected Wayne not to try to buy stock in his own. That would make his share minor
- the other members of the board were not as invested, they could be happy that the public market would made their stocks easier to be sold when needed.
The bottom line is that it would be more difficult to get an agreement of the holders of half the stock to dismiss him as CEO, since small investors sheldom have the opportunity/interest/information to vote.
For example, imagine that pre-public there where 5 investors with 20% share each, and that a 50.1% share is needed to dismiss Beale. It would only three of them to agree to overrule and/or dismiss Beale.
Now, imagine that after going public, the same 5 investor have 12% each (the rest going to small investors or Beale). Unless all 5 of them agree on something, Beale's will be the one directing the enterprise.
What Bruce did was to buy (through 3rd parties so it would not be noticed) enough of the new stock to dismiss Beale.
About from where did the money come, an important (unasked) issue is which was Wayne's control before. If it was 49%, it should not have been that difficult to get to 50.1%. Anyway, there are a couple options:
cash, as you noted, is not very likely since most of it would be invested
selling investments in other enterprises (after all, it is usually advised to diversify your investments)
loans (either on the stock being bought or other of Wayne's properties)
given the Wayne connections, he could also have convinced a group of millionaires to buy the stock and let him represent them.