This is puzzling to most people, because it is hard to imagine any number having a negative square. This isn't the case.
Imaginary numbers do exist.
Despite their name, they are not really imaginary at all. Before discussing why imaginary numbers exist, it's helpful to think about why we're even asking the question. One has to come to terms with the things that seem so puzzling and confusing about this concept and see that they are not really so unreasonable after all, before one can move on to accept the existence of imaginary numbers. Having done that, we can move on to seeing why they exist, and what relevance they have.