Most likely, the logs consist of what IPs are leased to what users, when the connections start and end, and what IPs those users are connecting from. A VPN company may keep the logs for something like 2 days.

Let's say you torrent something while connected to a VPN and one of the peers in the torrent pool is actually a DMCA agent associated with IP-Echelon. The DMCA agent will record the IP address you have at the time and generate a DMCA notice. It will then look up who owns the IP address to determine where to send the DMCA notice. When the VPN company receives the DMCA notice, it will use the logs to determine who was leasing the IP address at the time in question. If the logs no longer exist, the notice effectively gets tossed because the VPN company has no way of knowing what account was downloading the torrent. But if the notice was sent quickly enough for the logs to still exist, the VPN company will forward the DMCA notice to the user that was using the IP at the time. In that case, it will work the same way as a normal ISP. You'll probably get a warning with something like a 3 strike policy. In such a case, the VPN will cut your VPN service on the third strike.

Presumably, it could work the same way for anything. I used to work for a VPN company a decade ago, and this was pretty much the industry standard. It, like all VPN companies, advertised itself as having no logging.