Don't Understand Rooting

That link was not bad, but did not answer the question clearly, like the OP is looking for. Anyone can feel free to correct me here, but here goes. Android is more like a mobile version of Linux, for the operating system. As such, when the phone boots up, it needs to load a user profile. By default, it loads a normal user account, with no admin rights. As such, we can not load software in ways we may like, remove ones we do not use, and basically, do most any function that needs the higher access rights. So "rooting" is the term used to take the default account, and grant it the rights of an admin.

Hope that makes sense. If you use Windows on a computer, by default you have an admin account. If you make an extra one for your kids, it defaults as a standard user. So same basic idea, just in reverse of what happens on Android for us. Now as stated in the link above, when you root your phone, you do have access to things that can and will render the phone useless. Depending on what you want to do, this may be the wrong thing to do.

That explanation if pretty good.  Basically, in Linux terminology (And Android is a form of Linux) the "root user" is the default user that has access to all the system files which are normally protected.  The root user is also called a "super user".  Normally, system files are protected to prevent people from unintentionally screwing up their phones.  So keep in mind, its not for everyone.  But if you are careful, do the research, and only change what is known to work, then rooting can open up very cool options.  Some of these include:

1)  The ability to remove AT&T bloatware permanently.  Another mod defeats the upload speed throttle that ATT has on the Inspire.

2)  Change the system font, flash new themes, and change other aesthetics.

3)  The ability to flash (install) custom (non-official) ROMs.  You can get the latest version of Android (currently 2.3 Gingerbread) on your phone long before ATT releases it.  You can install custom ROMs created by the user community with highly modified aesthetics and tons of packed-in functionality. 

4)  You can also flash custom kernels, which allow you to overclock the phone's CPU, and improve performance.

5)  Some very useful market apps require root, such as ShootMe (for taking screenshots), Titanium Backup (for backing up applications and app data), ROM Manager (backing up the whole phone, or loading custom ROMs) and Ad Free (kills ads in many ad-supported free apps).

That's some of the key benefits of rooting, and there are many others.  Keep in mind that any such modification to your phone has a tiny risk of permanent damage to your phone, rendering it a useless brick.  Rooting may also void your warranty.  So again, its not for everyone. But if you follow the root instruction very carefully, do tons of research before you start, then you will be fine, and the chance of bricking your phone is very small.  If you want to learn more, Google XDA forums, and find the sub forum for phone, and read up. 

The files inside your phone are locked. Meaning they cannot be modified. When you "root" your phone you gain access to those files and have permission to modify them all as you wish. These files contain all possible parameters that need to be modified to make all changes to phone (if you know a little about windows systtem, the Windows registry allowed you to do all of this changes). For example, to force the phone to roam, to modify the state of the GPS, Wifi, Bluetooth as ou wish, etc. 

Also, when you completely root a phone, you can open the bootloader, and install a custom ROM or Kernel. A custom ROM is nothing more than saying I have a Dell laptop that came with Win 7 pre-installed, but I formated it, and reinstalled Win 7 the way I like it with no bloatware, etc. A kernel is the interface between the hardware and the software. There are a lot of modifications that can be done such as overclocking, undervolting to improve battery life, and lots of other stuff. Modified Kernels are looking to improve performance. 

That is the general idea behind rooting. You are gaining access to all files which allows you to modify the software as you wish. Hope it helps.