A company's management cares about share price (and sometimes dividends) because that's what the owners of the company (shareholders) care about. If the directors of a publicly traded company don't care about share price, then the shareholders will vote in new management that does.

At the same time, share price isn't something management can directly change. They do so by running the business well so that it generates profits and growth.

So the question "why would a company's management ever care about share price?" can be answered "because it's usually an indicator of whether or not they're doing a good job and provides job security."