The Router Operator role is assigned to members of the community that are operating a node. Routers are the liquidity providers of the Connext network. Their main purpose is to relay funds and data across chains. In exchange, they receive fees for their services. A Connext router is a node that automatically forwards in-channel transfers sent to it. For a crosschain transaction, this means that the Connext router is a liquidity provider that transfers funds in-channel on chain B in return for funds on chain A.
To learn more about routers, have a look at: What are Router Operators?