The details of this are unseemly and sketchy. This is what happens when you have liars conducting investigations into the alleged misconduct of others. How can you take the word of Project Veritas over anyone else's? They freely admit that they misrepresent themselves and lie to people during their investigations. If the FBI was caught doing the same thing, your heads would explode.
Someone with Project Veritas did call Ashley Biden using a fake name, and claiming that they wanted to return the diary to her. Project Veritas claims that during the phone call, Ashley Biden gave them enough information to conclude the diary was hers. It was actually a so-called "whistleblower" with Project Veritas who handed over the diary to National File, which included passages concerning sexual abuse... but Ashley Biden has not confirmed that she wrote those passages. Nobody has.
Here is an article from Salon which gives an overview of the events :
Trump supporter shopped Ashley Biden diary at fundraiser: report
This article also doesn't address the contents of the diary at all.