Because they are useful for describing certain things. For example, they are heavily used in electrical engineering and signal processing, because you want to work in two dimensions, and imaginary numbers are a convenient way to do that.

But the question is itself nonsensical. Nothing man-made needs to do exist. Negative numbers don't need to exist, you could just say there's no solution. Fractions don't need to exist, you could just say there's no solution. Natural numbers don't need to exist, you could just not count things. Language doesn't need to exist, we could just not speak, plenty of animals get on with their lives just fine without a language.

People really get hung up on the word "imaginary", and it drives them crazy, but it's just a name. Real numbers aren't any more or less real than imaginary numbers. Natural numbers aren't more or less natural than integers. Rational numbers don't have a mental capacity, and irrational numbers don't run around making dumb decisions. A turing machine isn't actually a machine with cogwheels inside. They are all made up, and they are all just names we gave them.