Owners of copyrighted work typically demonstrate that they are the rightful owners of the copyrighted work by introducing the copyright registration as evidence. As it was mentioned in an earlier post, copyright registration within five years of first publication creates a legal presumption of ownership and validity. While copyright registration does not prove ownership of copyright, this legal presumption of ownership and validity constitutes a prima facie proof (on first appearance). Consequently, in cases of infringement the court will presume that the ownership of the copyright and the registration is valid and will shift the burden of proof to the defendant to disprove by showing with sufficient evidence the falsity or invalidity of the allegation.

A third party may also bring a suit provided the owner of the copyrighted work has assigned (i.e. transferred) the rights to this party. In these cases, the party can introduce the transfer of copyright ownership written documentation as evidence to "prove" ownership.

References:
[1] Stim, R. "Intellectual Property. Patents, Trademarks, and Copyrights" West Legal Studies.
[2] Black's Law Dictionary 5th ed., (West Publishing, 1979).
[3] Copyright Act of 1976