First of all, the price of a company's shares is often used as an indication of the overall strength and health of a company. In general, if a company's share price has continued to climb over time, the company and its management are considered to be doing a good job. If everyone is happy and the company is doing well, as reflected by its share price, then management likely will see a raise and there is decreased risk that they will be fired. The management of a company is at a great risk of being removed from the company if they are unable to generate returns for investors. This is done by the Board of Directors, who are elected by the shareholders, and are responsible for the hiring and firing of the executives. If the share price lags, management likely will be removed.
Compensation is another important reason for the management of a company to keep the stock price as high as possible. Most executives in a company receive part of their compensation with stock options, which gives the manager the right to purchase shares in the company at a set price. In general, the price that these options are set at are based on the most recent price of the company's shares when the option is granted to the manager. Over time, for this option to gain in value and increase the benefit for the manager, the price of the security must rise. This is why the issuance of stock options to managers is considered a good way to align the interests of the executives and the shareholders, as both will want to see share prices rise over time.
There are other reasons that a corporation is concerned with its price including the prevention of a takeover. When a company sees its share price fall the likelihood of a takeover increases as the company is relatively cheaper. If a takeover occurs, the management of the company is often let go, which again goes back to management protecting its own interests as no one wants to be fired.