Okay, hosting...that basically means someone's computer has to churn out all of this stuff. Simple Machines Forums, this is just software that a computer is running. Someone else's computer. There are two categories, desktop computer and webserver. You use a desktop computer. The server is what your desktop computer connects to. When you browse a website, someone's computer, a webserver, is processing the requests your computer sends to it. Then it spits back some data, and your computer processes it.
Hosting: the forums software is written in PHP, I'm pretty sure. That is a programming language, many sets of instructions. That's why you see php in the url (that's in the address bar, like
http://www.google.com). The SMF forums stores all the information (usernames, passwords, forums posts, etc.) into a database, which is an organized collection of data, usually alot of data. In order for all of this to work, someone has to have the database on a server, the php programs (SMF forums) on the server, the images etc. on the server, and an operating system, the program that controls the whole server and allows all this complicated stuff to work. Last of all, we need regular ole desktop computers to access the servers, else the servers are useless, as they serve no purpose.
So, smfforfreee3 is hosting (in other words it owns it and lets you access it or its services) the os, the database, the forums software, the images, the security, pretty much everything. They put ads on the site, and in exchange, you get to sign up for free which no hassle (i.e. you don't need your own server and don't have to set much up).
Tell me if this is unclear or what part I need to explain more.