If I understand the problem, I think this is the correct advice. I've never had the problem at an airport; but have once or twice with hotel wifi.

In my untechnical opinion, it seems to have something to do with how Safari responds (or does not respond) to the network's attempt to "hijack" your first page request. You open your browser, then no matter where your normal home page is, the wifi network / proxy server will (attempt to) redirect your browser to their welcome page (where you pay and/or agree to terms).

It seems like Safari is not cooperative with this kind of electronic "involuntary reroute" (see, I made a funny!) ... maybe it's actually a security measure in Safari, I don't know.