A company's performance is measured by its stock price. If the stock is in the shitter, the sharholders can vote in new Directors or the Directors can fire the CEO. Since many big executives have options and compensation tied to the stock price, it means a lot to them where the stock price is. So first and foremost, the jobs of the Directors (main ones at least) and the CEO is linked to stock performance. A company uses its stock as currency sometimes to make acquisitions so this could be important to some companies. Many companies do subsequent offeringsand therefore want higher stock prices to raise more capital down the road. A dropping stock price is usually a reflection of the market's perception of the company and its products. So for simple pride and ego, a company wants the stock to go higher and for people to want to own shares in the company. Public image is very important. If you IPO at $40 and the market sells off the stock to $10 it does not matter if you already got your money from the IPO, it will have a negative affect on your company. Imagine what lenders will think, I assume they will want to look closer and see if something is wrong. Also imagine the IB that floated the stock, they will look really bad getting people to sign up for this stock only to have it tank real hard. Stock price affects the company in many ways. If your job was tied to stock price you would spend all your time making sure the company was growing and that the stock price was also growing.
The full answer could be long and winded. However, it can all be summarised with one word : GOODWILL. Basically, the difference between market value and accounting value ( book value ). Once the company goes public, this number is all the company is constantly being rated on. The more goodwill ( intrinsic value ) "created", the better for everyone involved. Managers take all the responsibility and fat cheques, shareholders become happier, etc. If the goodwill is perceived to be too low i.e. the company is undervalued, then all the vultures will come out of closet ( corporate raiders, hedge funds, direct and indirect competitors, etc ) wanting a piece of it and threatening the independence and integrity of the company. Hence, the directors will always strive for a high market value. This is what they are paid for. On the other side, if the goodwill is "too high", well ... that story is better left to speculators and short sellers. Hope it helps.
Price at IPO is usually the book value. Stock market value may be greater (usually and hopefully) or less (companies about to go broke). The main stockholders (those that hold massive quantities of their shares) exercise their voting rights to place/remove directors. Capable directors get the company to make profits, which usually translates into higher stock prices. So inept directors would be fired by the shareholders. Actually all shareholders have voting rights but many people have so few shares that they don't take the hassle of voting, given their small power.