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 servers that used to be located in the Network Access Points (NAP) 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.
A route server is used most frequently by network operations engineers who are trying to determine the cause of connectivity failures their customers are experiencing between two endpoints. Typically, the network is up, both sites are reachable from the NOC engineer's location, but the customer can't 'get there'. A route server gives a view into other provider's networks so that the NOC engineer can trace the problem to a bad route advertisement, aggregation or prepending errors.