> You can owe the mob 5 sheep. Which means that if you want to have 2 sheep, you'd better buy 2 sheep so you can pay your debts before you start farming wool.

Your math is wrong. If you owe the mob 5 sheep and you want to have two sheep, you're going to need to buy 8 sheep (5 for the mob, 1 for the "big" and 2 for you).

But you still do not have -5 sheep, just a debt of 5. That is conceptually not exactly the same. Sure, you can now define(!) negative numbers as debts, and that's okay. This would be one formal way to extend the natural numbers to the integers. Similarly one can extend further and further if careful.*

But exact numbers, irrational ones in particular, are already esoteric in real life. No fence ever will have exactly length pi. No diagonal of a square with side 1 truly has length sqrt(2), however precise you drew it. And maybe that third of a pizza was actually slightly less or more (but that one can be done, if we get down to counting atoms).

So what we do is to accept that those numbers mostly exist conceptually and abstractly. But if pi and sqrt(2) are fine, why not i? We artificially added the circumference of a circle and a solution of x² = 2, why not also x² = -1? And as mathematicians realized this is maybe where we can stop: every (non-constant) polynomial equation has already a solution in the complex numbers (they are algebraically closed), and every limit (such as pi as an infinite sum) that should exist actually exists (they are complete).

It also has applications in real life, and you don't need to go to quantum mechanics for that: The laws of electricity for DC extend neatly into those of AC. But only if you treat capacitors and coils as resistors of imaginary(!) "resistance". Hence like pi being the best way to deal with a fence of arbitrary precision, i works really well to deal with currents.

There are more abstract reasons in mathematics as well. As a simple example (as going into the true applications would go way beyond ELI5 or ELI18):

What is sin(0°) + sin(1°) + sin(2°) + sin(3°) + ... + sin(179°) + sin(180°) ?

Complex arithmetic tells you that sin(x) = ( e^ix - e^-ix ) / 2i, with x in radians. Using that and the geometric series

1 + a + a² + a³ + ... + a^n = ( 1-a^n+1 ) / (1-a)

will lead to the result; details are left to the reader ;-) .

tl;dr: they just work and make life easier, so why not use them?

*: The more common one is to work with pairs (a,b) of naturals, which we treat as if it were the number a-b: we consider (a,b) equal to (c,d) if and only if a+d = b+c (note how this only involves natural numbers now), similar to how "a/b" and "c/d" are the same if and only if a·d = b·c. And so on...