Unless the background check has the information the dealer is in fact taking the word of the buyer. However, as others are pointing out here, lying on the form makes the purchase illegal and the person illegally in posession.
The MSM's drooling reporters are loving to be able to report that he was able to legally buy the weapons. But if he lied on the form, he was in criminal possession from the start.
Technically, he committed a felony by lying on the form, even before he took possession - so he was a felon in possession at that point.