This is an interesting topic. When you think of a server you tend to think of a single machine running a single program. Farming is the idea of multiple machines running multiple programs that all act together to appear to be one machine/program.
Why farming? Two reasons with equal importance. Scalability and Reliability.
Scalability is the idea of allowing something to handle many many more connections/transactions at the same time. A typcial web server can handle say 1000 connections a second, but 10 web servers acting together can provide 10,000 connections per second.
Reliability is the idea that even if a piece of the system goes down, the system does not. There is no single point of failure, and there is built in redundancy to ensure that something is there to pick up the slack if something goes down. In the web server example, if one web server goes down you could not send any connections to it until it is back up, thus ensuring that your web site is always up.
Both of these ideas are crucial to a successful Farming strategy. Each wants to push the design into a different direction, but they can be reconciled into a single design that can scale and is reliable at the same time.