[Rondom]

Since you are connected to their network they know your MAC.

What they will likely do is take your IP and look to which MAC their DHCP-Server has assigned that IP to. There are other possibilities though, but they are more complicated (using ARP-cache or requests, doing a raw packet dump at the webserver).

If you want learn more, I think you could read a bit about the different layers first, and then read about how ethernet, IP/ARP and DHCP work. There are likely no "headers" (I assume you mean HTTP-headers) they inspect.