Economists care about long term trends in economic growth because only economic growth can bring about improved standards of living in a country. People want to enjoy a rising standard of living because it makes life generally more enjoyable. Governments want to enable their people to enjoy this so that the people will be happy and content with their system of government. Therefore, economists care about these trends.
Without economic growth, there can be no improvements in the standard of living. In order for standards of living to rise, there must be more production per person in an economy. As more production per capita happens, it means that the average person has more money available to them and more things that they can buy with that money. When people are able to buy more things (at least to a certain point), the quality of their life generally improves.
Economists need to care about growth, then, because people and governments care about growth. Growth brings better standards of living and that is what people want.