What the authors are trying to discuss is how to think about a company which has retained earnings, in terms of the 'cost of financing'.
Because you want this to non-corporate finance people, first some definitions:
(1) Retained earnings: these are accrued profits from previous years, that haven't yet been paid out as dividends. ie: if Ford had net income of $1B last year, and paid out $300M in dividends to shareholders, it would have $700M increase in retained earnings. This would be added to all the opening retained earnings from previous years.
In most jurisdictions, corporate law generally prevents a company from paying out dividends greater than retained earnings. This is basically done so that a corporate owner can't cheat anyone it owes debt to, by borrowing funds and then paying those funds out as dividends and then dissolving the company. You could say that retained earnings basically represents cash that the company has earned, but hasn't yet paid out to its shareholders.
Keep in mind that decisions made by 'the Corporation' are really decisions made by the Board of Directors [who are elected by the shareholders, and who hires the CEO and ultimately decides on serious matters such as dividend payment policy]. Therefore, you can assume, which these authors have done, that the Corporation represents the shareholders, and that the shareholders have effectively decided that they don't want dividends now. Probably there is an investment strategy that means the money would be used by the corporation to earn more value in the future.
(2) Cost of financing (also called 'cost of capital', like in the title of that book): This refers to the implicit annual expense for the company to be able to afford all of its assets. In essence, if your business can't earn enough to cover your cost of financing, then it is actually losing money overall, even if it may appear profitable on paper. In simple terms, if you own a rental property for $1M, and earn $70k in rent each year but pay $100k in financing costs [like mortgage interest], then you lose $30k per year.
There are, for simplicity, 2 types of financing:
(a) Debt Financing: This one is the simpler to understand. If you opened a business, and wanted to buy a factory for $10M, and buy your opening inventory supplies for another $2M, and then have a few months of payroll saved in cash for $1M, you would need $13M. If you went to the bank, first assume they would just loan you $13M, at an interest rate of 7%. Your cost of financing, AKA your cost of capital, would simply be 7%.
We can now take your 7% cost of capital, and say that any asset earning less than 7% should be cut from your business. So if half your factory produces widgets, and widgets make you $500k in cash profit each year, using $6.5M of your total $13M in assets, then you could say they are earning you a 500/6500 = 7.69% rate of return. This 7.69% is higher than your 7% in financing charges, so in the end, even after paying interest costs, your business is profitable. If your other half of the factory produces stegdiw's, which only make $100k / year, they earn less than your cost of capital, meaning that half of the business is losing you money after interest costs.
This is why cost of capital is sometimes used as 'required rate of return', for the business, meaning it is the minimum amount that any group of assets should earn to be profitable.
(b) Equity Financing: This is the charge that gets assumed, that gets allocated for equity holders [ie: shareholders]. Shareholders don't get mandatory interest payments, but they have a right to all net proceeds from the company, and they get those net proceeds in the form of dividends, or else in as net cash when the company liquidates. Remember though that debt holders take on more risk than equity holders, because debt holders must always be paid interest annually, and also because debt holders get paid first in a bankruptcy situation. Equity therefore has a financing charge far higher than debt, because equity holders take on more risk than debt holders.
In the example above, assume the bank doesn't want to lend you the full $13M, because your business would be too risky [one bad year, and you might not be able to cover your interest payments]. So, they agree to loan you $10M, and say you must put in $3M of your own funds. Now, your interest payments have reduced to 700k / year, instead of 910k. Great! Except now, you need to consider how much YOU need to be compensated, for the risk of putting your $3M into the business [after all, you could have invested that money somewhere else!]. Because you took on more risk than the bank, you need more compensation - let's assume fair compensation would be 13% [in reality, calculating this is quite complex, especially for small businesses, but 13% is probably in the ball park of what would be 'fair', if the bank thinks a 7% interest rate is fair].
So in addition to needing $700k / year in interest payments, you need to accrue earnings to equity holders [you] of $390k. So your total combined cost of capital is (700k + 390k) / 13M = 8.38%. Your cost of financing increased, because you have a higher proportion of equity [now 23% equity], and equity is more expensive than debt.
So how does this all relate to your question?
Assume your business now earns $4M. You pay interest of $700k, and you even pay $390k in dividends, and are left with $2.91M in cash after all cash outflows. Great! You earned higher than your required rate of return. If you keep the cash in the business, though, you now have debt of the same $10M, but equity of $3M + $2.91M [because retained earnings is another form of equity]. So your cost of capital is 700k interest + [5.91M Equity * 13% = $768k cost of equity] = 1,468k annually, over total assets of [13M + 2.91 increase in assets from first year profits = 15.91M], = 9.2%.
Your cost of capital has actually increased further, because a higher proportion of your company is effectively funded by equity! This makes sense, because per the assumption in your question, the corporation is acting in the best interest of the shareholders, and the shareholders kept their 2.91M retained earnings in the company, instead of declaring a larger dividend. They could have invested that 2.91M somewhere else, but they decided not to, so it should earn them the same value within the business, as it could elsewhere. Since we are assuming 13% is a fair compensation for the high risk involved in this particular business, they must earn that 13% on the total 5.91M equity invested in the business.
Therefore, for purposes of calculating the cost of capital, you must include the value of common shares originally invested, + the value of all accumulated retained earnings which are still ultimately value attributable to the shareholders.