A php "program" is really just a file or set of files.
Go to the SMF website and get the latest version (full install, ZIP I guess). In the zip you'll see a bunch of files and folders.
Put these up onto your webspace, wherever you want your forum to be. So for RMRK.net, they're in the folder at the root of the domain. If I put them in a folder called "forums", you would be visiting these forums at "
http://rmrk.net/forums/"
When you go there it should show the installer. SMF works with a mySQL database, which you'll need to create for it in your host's control panel. Can't help you much there unless they use cPanel (90%+ of hosts, including me do), but if it is cPanel then it's easy enough for you to do it by yourself