What is a callback function?
A callback is an idea of passing a function as a parameter to another function and have this one invoked once the process has completed.
If you get the concept of callback through awesome answers above, I recommend you should learn the background of its idea.
"What made them(Computer-Scientists) develop callback?" You might learn a problem, which is blocking.(especially blocking UI) And callback is not the only solution to it. There are a lot of other solutions(ex: Thread, Futures, Promises...).