Possible kinds of evidence
As explained in the comments by Nate Eldredge:
There only really is a general answer. You present evidence and see
if a jury believes you. There's no specific set of evidence that
they're required to accept or reject; it's always case-by-case.
There isn't much that is inherently legal about how you prove once case v. another. Proof of facts depends upon what information you can get and basic logic and emotional impact. The rules of evidence rule out some kinds of evidence but they don't tell you how to affirmatively prove particular things.
Testimony
The most obvious way to do so is to get a statement from the person who allegedly created the work under oath, or from some other person with personal knowledge of the facts, or from someone who heard someone else with personal knowledge of the facts say so. Cross-examination of this testimony might also be required to clarify points that are vague in that testimony.
There is a common misconception that testimony from a party to a dispute isn't evidence shown by statements like "What proof do you have?" But, in fact, it is the most common kind of evidence, is generally admissible, and is routinely used to prove facts in legal disputes.
Propensity evidence
Propensity evidence might be admissible to show an established modus operandi of this particular person generating content with AI and then claiming they made it (or of not doing so), or to impeach the credibility of a sworn statement from someone providing evidence about the source of the work.
Someone who has been caught previously submitting AI-generated works to contexts and in copyright applications is going to have a harder time proving that they made the work themselves than someone who has a long history of painstaking making original works without AI assistance.
Forensic review of computer files and browsing histories
One could subpoena the Internet browsing histories of the suspected AI-generator and the metadata of the file being advanced as human created, to show opportunity to use AI, and a timeline that favors an AI or non-AI source.
For example, if a large complicated work was created in five minutes, shortly after the browsing history shows access to an AI-generator website, this would be strong evidence that it was created with AI, while the absence of access to an AI-generator website and scores of versions of the same work over a period of months from an application capable of being used to make a work like that which was present on the person's computer, would be strong evidence that the work was created by a person.
Review of public output from AI-generating websites
Some AI-generating websites publish to the general public works created using it, so a search of those websites for that kind of republication generated with user input might be possible.
If the work submitted shows up in the AI-generating website's public output, it is almost surely AI-generated.
Expert testimony
Another way to prove this would be to get expert witness testimony from experts in distinguishing AI-produced works from non-AI produced works (a growing cottage industry). Short of that, there is software that is designed for the purposes of distinguishing the two, although even if this is admissible as evidence in court, it would probably be given much less weight than expert witness testimony.
The details of how this is done by expert witnesses is beyond the scope of Law.SE and calls for information technology and artificial intelligence expertise, rather than legal knowledge. But the answer by Barmar explores some of them in a cursory way to illustrate the kind of testimony that expert witnesses might provide.
What counts as AI-generated for copyright purposes?
None of this engages with the legally hard question of where AI-generation that is not entitled to copyright ends, and where the use of computer generated graphics or editing software in the course of preparing a human generated work begins. This is another reason that expert testimony may be required.
Does Photoshopping a picture cross the line? Is telling a computer to color in everything between certain lines a certain shade of blue AI-generation? Is hand drawing six images in a twenty-four frame-per-second animation and asking a computer to interpolate between them for the other 18 frames AI-generation? Is using a 3D-model of a common part of the set as a starting point for drawing something in an animation or manga AI-generation? Is having a computer do shading from the direction of a light source identified by the creator AI-generation? What about using an AI-generator to generate a trailer or storyboard to used to inspire a longer or more complete work that is created the old fashioned way?
Is following AI advice in a grammar editor to break up run on sentences and correct grammatical and spelling mistakes and avoid using words considered derogatory AI-generation? Is using an AI-search to find references and quotes to consider using in research paper AI-generated if the text that utilizes those references and quotes is human written?
Many of the gray area questions remain unresolved legally.