I think what people who are saying that are referring to is the collectively bargained minimum salary. If the NFL is forced to operate with no CBA, there will be no minimum salary, and many players who would have made the minimum for their experience level could end up getting much less money. There's no recent evidence for it, of course, because the NFL has had a minimum salary for decades. But imagine an NFL with no salary cap, no salary floor, no minimum salary and completely unrestricted free agency. The distribution of money among the players likely would be drastically different, and the total amount they collectively receive might be even less than it is now.
Knowing that they will have to outbid every other team for the best players, even their own, will teams still want to pay the worst player on the team more than $320,000 per year (for a rookie, more for a veteran) to be inactive each week? Or will they be able to find someone who is more desperate to do it for much less? More likely, the borderline players who would have gotten the minimum of $405,000, $480,000, $545,000, etc., under the old CBA would accept less because teams won't have any reason to offer them that much anymore. There's no bidding war for them -- they're just looking for any team that will sign them.
If there was open bidding for Peyton Manning, Tom Brady, Aaron Rodgers, DeMarcus Ware, etc., how much higher would their salaries be than they are now? And if the Colts had to pay twice as much for Peyton Manning, wouldn't they likely spend that much less on their other players?
Now, how extreme the salary spread between the elite and the dregs would become is unknown, but there's no question that it would increase.