My professor claims that a company want the value of its stocks will be as high as possible. Why should the company care? The company "minds its own business" and make its money. Why should it cares if the value of its stocks is \$25 or \$300?
The board of directors is supposed to represent (or work for the interests of) share owners. Share owners prefer higher prices if and when they want to sell their shares. The board chooses the CEO and will retain the CEO if he helps share owners realize their preferences.
Objectively, the value of a share is the discounted stream of dividends. That is, an investor buys a share because she expects the firm to distribute dividends (which are at least positively correlated to profits). Different traders may have different beliefs regarding these dividends and their beliefs are aggregated in a price "on the market".
The board of directors is elected by the shareholders. The shareholders either want to keep the stock and continuously receive large dividends or they want to sell at a high price. If you want to keep your job in the board, you want to make shareholders happy and you do this by running a successful business in the sense of high profits $\rightarrow$ high dividends $\rightarrow$ high stock prices $\rightarrow$ happy investors $\rightarrow$ job security and high salary for the CEO (you).
The idea behind this is shareholder value.