NARA has not exactly explained in public how they concluded some documents were missing. One can imagine that they cross-referenced things and so forth. But the DOJ said NARA did ultimately complain to them.
In February, Archivist David Ferriero testified before Congress that his agency began talking with Trump's people right after they left office and that the Trump camp had already returned 15 boxes of documents to the Archives. Ferriero said that in those materials, the Archives discovered items "marked as classified national security information," unleashing further inquiries as to whether Trump continued to possess classified material. [...]
[Trump] has previously said that he was returning any official records to the Archives, labeling any confusion in the matter as "an ordinary and routine process to ensure the preservation of my legacy and in accordance with the Presidential Records Act." He also claimed the Archives "did not 'find' anything" in what he had already been returned, suggesting that there was nothing sensitive. He said the documents had inadvertently shipped to Florida during the six-hour transition period in which his belongings were moved.
According to the Justice Department source, the Archives saw things differently, believing that the former White House was stonewalling and continued to possess unauthorized material. Earlier this year, they asked the Justice Department to investigate.
In late April [...] a federal grand jury began deliberating whether there was a violation of the Presidential Records Act or whether President Trump unlawfully possessed national security information. Through the grand jury process, the National Archives provided federal prosecutors with copies of the documents received from former President Trump in January 2022. The grand jury concluded that there had been a violation of the law, according to the Justice Department source. [...]
Put in place after Watergate to avoid the abuses of the Nixon administration, the [PRA] law imposes strict penalties for failure to comply. "Whoever, having the custody of any such record, proceeding, map, book, document, paper, or other thing, willfully and unlawfully conceals, removes, mutilates, obliterates, falsifies, or destroys the same, shall be fined" $2,000, up to three years in prison or "shall forfeit his office and be disqualified from holding any office under the United States."
As grand jury proceedings are not public by default in the US, this is probably as good as it gets for now. (It's also worth noting that e.g. the Nixon grand jury records were ultimately made although this was certainly after some time, and a on a quick look, there are still some redactions in there.)
N.B. apparently the procedure is that Trump (or any former president) gets to designate their representatives for handling the turn over of such documents/records (to NARA). Trump apparently designated "John Solomon, a right-wing journalist" among others. Said journalist was also the first to report that a grand jury subpoena was received by those designated Trump representatives (per previous link). The other designate was apparently Kash P. Patel, a lawyer. According to the Washington Post, designating Solomon as a representative was in itself problematic, because "Solomon: as a news writer, he would not have had clearance to view those documents" (if any were indeed classified). Solomon denies having seen any documents though.
Oh, and Trump says none were classified... because he declassified them.
On the day the search warrant was unsealed, Trump put forth a new explanation about why he kept highly classified documents — and it’s likely to be at the center of his legal defense moving forward.
Trump’s office provided a statement to John Solomon, the conservative journalist who is one of the former president’s authorized representatives to the National Archives. The statement said Trump regularly took classified material to Mar-a-Lago, and that he had issued a never-before-revealed “standing order” that documents removed in this fashion “were deemed to be declassified.”
“The power to classify and declassify documents rests solely with the President of the United States,” the statement read. “The idea that some paper-pushing bureaucrat, with classification authority delegated BY THE PRESIDENT, needs to approve of declassification is absurd.”
I suspect this claim might be further challenged in court(s) in term of the procedure followed, e.g. who knew about the "standing order". Also, as discussed here if them documents are deemed declassified, that only absolves him and his staff/representatives of the mishandling classified material charges, but not necessarily of the other potential charges regarding (unclassified) presidential records handover to NARA (or rather for lack thereof). And Presidential Records Act aside, more obscurely, even not classified materials could potentially be considered in violation of the Espionage Act (that was actually invoked in the FBI warrant besides the PRA), because that law predates the modern classification system, so it doesn't reference it. I have no idea if any such cases were recorded though in recent times. That might make a good Q on Law SE.