After the initial public offering, the company can raise money by selling more stock (equity financing) or selling debt (e.g. borrowing money). If a company's stock price is high, they can raise money with equity financing on more favorable terms. When companies raise money with equity financing, they create new shares and dilute the existing shareholders, so the number of shares outstanding is not fixed.
Companies can also return money to shareholders by buying their own equity, and this is called a share repurchase. It's best for companies to repurchase their shared when their stock price is low, but "American companies have a terrible track record of buying their own shares high and selling them low."
The management of a company typically likes a rising stock price, so their stock options are more valuable and they can justify bigger pay packages.