IRC is the a global chat system, comprised of many individual hosts, all (eventually) linked. You usually don't run a part of IRC yourself, rather you use a client to connect to one of the hosts and thus gain access to the network and everyone on it.
An IRC bot is a piece of machine code that has its own logic (Eliza's logic, in your case), and is also connected to IRC and issues IRC commands that let it join channels, talk to other people/bots, received messages, etc.
In your case, you would simply need to run the bot, and let it know which IRC server to join (that the server
parameter in new Eliza({...
), what nick name to choose for itself, etc.
To run the bot just run node
, in any environment (your home PC? hosting?) with the contents of the sample file.