You cannot play Age of Empires II HD on LAN. To start a multiplayer match you require an internet connection.
The only way to have a match is to connect via the Lobby Browser, which requires you to be logged in with Steam. Once you have started a match, it will use the players IP addresses to determine how they are connected. If all players are on the same router, it will act as a LAN game, and not send data over the internet.
There are very, very, many threads on Steam supporting this.
Below, trying to start Multiplayer when not connected to internet:
And with internet: