I understand that if I accuse someone of copying my work that they could sue me. How can that happen?

Depending on what you accuse somebody of, the law allows the person accused to take action against you, for what is known as a “groundless threat”.  If you make your statements public, such as writing to suppliers or posting things on the internet, and if those statements are wrong but it still affects the reputation of the Defendant, they can sue you for unlawful interference in their business and for defamation. So, if you are going to make allegations of copying, make sure that those allegations are justified.