Role of complex number in electronics
It seems to me that you are puzzled by the term "imaginary number".

Complex numbers, and their "imaginary component" aren't more "imaginary" than "normal" numbers. They are a mathematical abstraction. Numbers, as we know them in math, don't exist in real life. 

If you are put off by my statement, think a bit about it and ask yourself what is, for example, the number 2. Can you really explain what is actually the number 2 in layman terms? 

It took mankind centuries to come up with a mathematically solid theory of natural numbers (20th century, Peano, for example - but it's not the only theory).

Other kind of numbers are more complex abstractions built on natural numbers. Integers, such as -2, are an extension of naturals. Rationals, such as 2.3, are an extension of integers. Real numbers (which comprise rationals and irrationals such as &pi;) are an abstraction built on rationals.

The point is, are those abstraction useful to us?

Complex numbers are just another abstraction built on reals.

Many of those abstractions have been invented by humans to solve some practical problem, even before they were formalized in a coherent mathematical theory.

Complex numbers are powerful, for example, because someone found a way to model electric circuits, under some assumptions, using that kind of numbers.

There are many more applications, as you correctly suspect. The answer to your "why do we use them?" question is "because they help us solve many problems".

There are many stranger entities in math that are used, in a sense, like numbers, and find application in many disciplines like physics and engineering: vectors, matrices, tensors. They all help us to model a part of the world we are studying and use a quantitative approach to solving some class of problems.

Bottom line: complex numbers are strange for you just because you are not accustomed to work with them, yet. For some amazonian tribes a negative integer would be as strange as a complex number and the reason is the same: they never learned how to use them.