Deriving and explaining the weighted cost of capital
If you are asking "Is the WACC the amount that the company expects to earn on the stocks and bonds that it holds.." then the answer is no. 

The WACC, in very simple terms, is the amount of money a company pays to obtain financing for projects. These types of financing are clearly listed in the wikipedia article and clearly extend beyond stocks and bonds issued by the company. 

A company can calculate the WACC to determine whether or not undertaking a project will be profitable. That is, a company can determine that it must earn some minimum return on a project before that project becomes profitable. 

You, as a consumer, possibly have experience with cost of capital. Consider the interest paid on a car loan. That interest is the cost you incur for financing the purchase of your vehicle by borrowing funds from a financial intermediary. 

As far as derivation goes - I'm not sure how to simplify these formulas much more. However, you can think simply about the first as the ratio of the interest paid out on sources of financing and the total market value of those sources of financing (so...some ration &lt; 1) and the second is a more simple case of the first. 

http://www.investopedia.com/walkthrough/corporate-finance/5/cost-capital/wacc.aspx

I think this website gives a decent walkthrough of the concept. Hope this helps some.