When you take calculus and differential equations (or Calc IV) Imaginary numbers (and eulers number as well) turn up there are are used in what are called "transforms".

The idea of a transform is: I have this disgustingly difficult calculus problem that is beyond my ability to do by hand. But...... I can transform it into an algebra problem, solve the algebra problem, then transform it back and get an answer that is useful. Yes, these transforms can get very messy on their own because you have to transform the entire algebra equation and take it with you.

The easiest analogy I can think of (and this is imperfect): You have a problem were you need to travel across the US continent, by foot. Walking 3000 miles kinda sucks though. So instead you arrange a plane ticket, fly in the plane, then get off and end up and the same place as if you had walked. There are some different tricky issues with the plane flight (like getting through airport security), but overall it is faster.

In this case, the plane is the transform. And if you have never seen a plane or used one, you look at it and say "what is that big aluminum can, I don't see any use for it"