I've worked with a number of captive portal systems and they all basically work the same way. The AP/controller intercepts http requests and redirects to the captive portal page with identifying information about the device (ip,mac,ssid,ap_mac,etc.). The captive portal http server shows the user a splash page to accept terms or enter a username/password or a credit card. Once the captive portal server decides the user should be allowed onto the network it needs to communicate that back to the wireless hardware which is done with the user's mac address.
Based on the requests it looks like they have some ads/trackers on the splash page that are getting requests with a referer set to the original splash page url (which includes the client mac address). A no-referrer meta tag or an intermediate redirect would prevent this from happening.