It might sound a little silly, but archaeologists can find out a lot about people by looking through their trash.
People throw away things because those things aren't important or because the people have too much of those things already. By sifting through the garbage pile of a civilization, archaeologists can find out what was important to those people (or what they had too much of).
Why is this important? Sometimes, garbage is all that's left of a people. Especially if that people has been conquered by others, the buildings, tools, and food was probably consumed or destroyed long ago. A people's trash, especially if it was also trash to the invaders, might be left alone, enabling archaeologists to discover more about a people who left few clues to what they liked and didn't like.
Did they wear certain clothes? What kinds of food did they eat? What kind of tools did they use or throw away? Answers to all these questions can be found by sifting through a midden.
It could also be possible to find out more about a conquered people by searching the midden of their conquerers. Some invaders, not really knowing what's valuable to the people they're conquering, might very well throw away things that are extremely valuable. It is left to archaeologists to find these things and piece together the life stories of people long since conquered.
One person's trash could be another person's treasure.