These instructions are obviously slanted towards a linux OS. With MSWindows it gets a little more fiddly. The computer has to have a mysql client installed in order for this to work. In MSWindows you get a client by downloading the mysql community server, doing a custom install, and selecting only the client for installation. You have to be careful to download the same version of the client as the RMySQL was compiled under. You have to go into MSWindows environment variables and create a MYSQL_HOME pointing to the place where the client is installed. Then you're ready to issue the library(RMySQL) command.
The firewall at Queen's University Belfast, which is hosting this database, allows access on a per computer basis. If you want access for R, contact Ron Reimer.