How saving and investment influence growth in the long run? Here are 4 points from my class.  


  • Because resources are scarce, devoting more resources to
  producing capital requires devoting fewer resources to producing
  goods and services for current consumption.
  
  • Consequently, the accumulation of capital involves a tradeoff.
  
  • When governments encourage saving and investment, they also
  encourage growth and in the long run this raises the standard of
  living.
  
  • A well-functioning and carefully regulated financial market, one
  that quickly and efficiently brings savings and investment together
  with minimal risk and in a transparent way, is a critical ingredient
  in the recipe for economic growth.


I can conclude that saving and investment are good for growth, but I cannot tell exactly why. Can someone make it clearer, maybe with a concrete example?
The result you are asking about follows more formally from the Solow model. It's definitely worth looking into. I won't go into the model specifics, since I believe you are after a simple example.

First, note that what economists refer to as capital is not really what most people mean when talking about capital. For us, capital is typically a production factor that is not labor. A good example is a machine. For agriculture it could also be something like fertilizer. These are all things that help you produce more and that is how you get growth from capital.

Suppose you are a farmer who can produce 10 potatoes a year currently. You sell those potatoes and then can decide to spend that money consuming (e.g. buying steaks) or you can invest it in "capital". If you spend the money on a tractor and some fertilizer, you can produce 20 potatoes next year. That is how investment leads to growth. That is also why you need to give up some consumption today in order to grow your production and have more consumption tomorrow.