How is an AAG listener implemented? Where does it run?
The Availability Group Listener (AGL) is part of the AG's Windows Server Failover Cluster (WSFC) Resource. It is a Virtual Network Name with n number of IP addresses. Being in a cluster resource it will only run on a single node of the WSFC and that node will be the primary node of the AG. It is not a thread within SQL Server. The WSFC handles the failover of the AG resource and the AG databases will not come online without the AG resource residing on its node of the WSFC. 

The listener just handles connection forwarding. When connecting to the read/write replica the connection will hit the listener and then the listener will tell the client to connect to the primary replica's IP address. In the case of read-only routing it will follow the below work flow.