Hotel Wi-Fi often has a captive network portal, where you are asked a username and a password.
Apple TV is not compatible with captive network portals, as it is meant for home networks, where the Wi-Fi only has a password requirement.
Either ask the hotel network admin for assistance,
or ask if it is okay to use your own (travel) router* in between their network and your devices. Then the travel router can connect to captive network portal of the hotel, and your devices can connect to the travel router like any home network.
* Other routers may work very well for this purpose too, but travel routers are designed for this setup, and thereby much easier.
** Having you own small network attached to the big network, allows for home network features such as AirPlay and Home Sharing, as well some form of security: keeping others out of “accidentally” accessing your Apple TV and other devices.