This is something that bugged me too. By right, a company shouldn't bother about its stock price/performance after it already raised the IPO right? So why do they bother?
Based on my research, these are some of the reasons:
1) A large percentage of the stock is sometimes held by the company's treasury or subsidiaries
2) A large percentage of the stock is sometimes held by senior managemenet - therefore, management has the incentive to ensure the stock performs well so that they have a high net worth
3) The market capitalization (total stocks outstanding x price per share) of a company does give it access to higher loans from banks
4) If a stock is undervalued relative to the intrinsic (i.e. fair) value of the company, the company can be subject to a hostile takeover attempt
Amount of stocks held by all investors multiplied by the stock price = the market capitalization of the stock. This is just an "imaginary value" and doesn't actually mean much since obviously the more the number of shares liquidated, the lower the share price will be. It definitely doesn't mean that the company is worth that much.
PS: Try to talk about this in the stock sub-forum, not this thread
A simple answer would be a company should care about its stock price because its investors care and if the investors are unhappy, they will kick the management out and hire a new team.