The browser does not know anything about the authentication. It is done on the outside: outgoing connections are intercepted. This can be done in several ways; usually, the WiFi router intercepts outgoing TCP connections, and, if the contents of the connection look like an HTTP request, the router serves back the 'pay me' page; all other connections are simply dropped. Once you have paid, the router lifts the restriction and lets TCP connections flow. There is nothing specific about WiFi here.
This kind of access control can be bypassed by using IP-over-DNS: since the router lets DNS requests pass (it intercepts only TCP connections), you can encapsulate arbitrary data traffic within fake DNS requests (provided that you control a DNS server on the outside, which does the reverse operation).
EAP, Radius... are about authentication protocols and their integration within the WiFi connection process. This is irrelevant to your airport situation: you connected without a password, hence, at the WiFi level, there is no authentication whatsoever.