A route server provides a look into the IP routing tables of the autonomous system in which the server resides. The concept of a route server has it's origins in the old Unix-based route serves that used to be located in the Network Access Points during the early days of the Internet. These Unix machines were configured with custom routing software ('routed', pronounced 'rout-dee'), designed specifically to make best-path calculations, and distribute a routing table to the routing devices forming the backbone of the Internet at these major peering points.
As custom routing hardware became more and more powerful (and cheaper), most NAP and CIX managers started setting up Cisco routers with open logins. This reqires less manhours and less work than many other methods. You can telnet to these routers and get a direct look at another network's routing table, and test connectivity.
Here's a short list of the more popular route-servers available to the public on the Internet. I find it suprising that very few of the major backbone providers do not provide such useful tools.