When you are asking about why they exist, I take it you mean why they were developed? Because if you're really asking about whether numbers exist, that becomes a philosophical and rather complicated question about our ontological commitments to mathematical entities.
They were first noticed possibly when mathematicians were solving quadratic polynomials, i.e. $ax^2+bx+c=0$. You'll quickly notice that sometimes we get solutions involving taking the square root of negative value. Mathematicians dismissed this as being absurd until they began to work on finding a formula for the roots of the general cubic polynomial, i.e. $ax^3+bx^2+cx+d=0$.
As for what they do, they have a lot of applications within and outside of mathematics. We're able to solve a lot of problems which appear to be firmly fixed in the real numbers using complex numbers. Within mathematics, this can be seen in geometry, calculus, etc. Outside of mathematics, it is extremely useful to physics and thus useful to engineering, particularly electrical engineering.
If our goal was to "get rid" of the negative, sure, multiplying by a negative number would get rid of it symbolically but then that changes our equation algebraically.