An IXP typically presents itself to it's members as an ethernet network. Smaller IXPs may be built out of simple Ethernet switches, big IXPs will likely use advanced features to present themselves as an Ethernet network while allowing higher performance and redundancy.
A router has a table of currently active routes. For each route the table normally holds both an interface and a next-hop IP address.
On ethernet networks the next hop IP addresses is translated to a MAC address through the ARP table (and if nessacery by making an arp request). This means that a router on an IXP can forward data to any other router connected to that IXP.
Route servers are not fundamentally required for an IXP, providers can and often do set up BGP sessions directly between their routers and larger providers often choose not to use the route servers to give themselves more control over their peering. The actual data path through the IXP remains the same whether the route was received from a route server or a direct BGP session.
The benefit of route servers is for members who want to peer as widely as possible. They can buy their port on the IXP, establish a single BGP session to the route server and immediately start exchanging traffic with like-minded peers without having to go through the hassle of making individual arrangements.