AOL was never in the privacy business, they were an ad company right from the start. So someone complains to the cops, the cops send a nastygram to AOL demanding information on such-and-such a screen name. If he actually had AOL internet service they'd have a name and billing records, which they'd then give the cops. If it was just AIM, they'd have the IP address. Which he was unlikely to be hiding 20 years ago, VPNs weren't really a thing. So they go one more step and send another nastygram to whatever residential IP has that address space, and demand do know who had it on date X. The ISP will have billing information that includes a real name and address, and bob's your uncle. Take these documents to a judge and he'll sign an arrest warrant for making death threats, go to the address and arrest the guy and charge him.
Would cops in 2006 have actually gone to the trouble? Maybe, maybe not, there was still something of a sense back then that being mean to people on the internet wasn't exactly a crime. But they might have.