There are a number of reasons to recycle waste. First, it is often profitable to do so in, even purely in terms of money. Aluminum and copper, for example, are quite valuable, because it is expensive to produce those metals in the first place. Why should we have to go out and refine vast quantities of ore at great cost to obtain metals which are already abundantly available in the form of used objects that we would otherwise throw away?
In addition, we as a society produce vast quantities of garbage, and it is difficult and expensive to find places to put it all; garbage dumps fill up and new ones have to be opened, and we are running out of room. There is also an environmental cost to filling the world with garbage and plundering all available resources to make even more garbage constantly. We are turning the whole world into a vast garbage heap, and we are not going to like living in it when we do.
Finally, the world only has a limited quantity of resources. The population of the world continues to increase, as does the level of industrialization in many nations, and the level of consumption rises accordingly. We really are going to run out of things. And if we can re-use what we have, we won't run out as quickly.