How does this work
Your VPN uses a server in Private Internet Access public network to be able to work. The fact that those addresses are used by PIA is well known (it's probably registered in the public WHOIS database too). It is probably known to whoever designed that WiFi security.
As soon as whatever routes packets in that café sees a packet directed to any of a largish number of uncool address pools, the originating device is disconnected for security reasons.
One way to do it is with a Linux router box with iptables, fail2ban, and a custom script to send the disconnect command to a compatible router (I think I know how to do that with a D-Link access point, as it has a Web interface that could be scripted via curl; but most APs are probably scriptable that way). Set a limit of, say, 20 packet attempts in a minute to trigger disconnect, and Bob's your uncle.
and how do I bypass it?
You need to find a service that offers VPN or proxy services, and yet is not blacklisted. Whether one exists, is more than I could say.
If the network uses a whitelist for allowed sites, and a blacklist for sites that trigger disconnection, you might find such a service and yet it will not work - you'll remain connected to the WiFi (not being blacklisted) but unable to navigate (not being whitelisted).
Frankly, moving your custom elsewhere or procuring a cellphone data plan with WiFi tethering both look more promising than trying to bruteforce access.