What is the idea behind a projection operator? What does it do? I know what a projection operator is, but I am unable to explain it in words without using mathematical symbols. Can anyone help me?

I don't need examples or the definition - I want to know why and how its need arose, and what is the idea behind it?
This is how I used to imagine projections:

If  a mouse:



gets run over by a steamroller:



It will look like this:



Now if it gets run over by a steamroller another time, it will still look like this:
A projection map $f: A \to B$ is something you often need to simplify the space $A$. It is build so that differences which occur in the space $A$ will often no longer occur anymore in the space $B$. Then, if you know enough about the map $f$ and find out a bit about the space $B$ (which is easier as their are less differences) then you may find out interesting things about $A$.