Well generally, they don't want root access. Root access allows you to actually look into the "root" of the folder, somewhat similar to being able to look into the C drive on the computer. They want you to be restricted to the "my documents" folder, if that makes any sense.
Root access allows things like sideloading (which ATT and I think another doesn't allow) which can lead to people downloading cracked apps (which isn't always the case if you go to XDA). You can also tether without the required plan. Of course, we don't all do this but it is a nice add on.
Aside from that, there are other applications that require root, such as setcpu which gives you complete control over the CPU speeds, allowing you to limit it to conceive battery life and things like that. You also can use applications like Titanium backup, which allows you to create replica backups of applications and their data, which is amazing for games and other programs.
Now with root you have access to things like ROMs and kernels. Roms in a sense would be the.... version of windows per say. For instance, the Samsung Galaxy phones over here in the states shipped with 2.1 and just now got 2.2. If you rooted the phone, you could install a custom rom with an actual version of 2.2 WITH an underlying kernel that allowed for overclocking and things of that nature.
The ROM was the update or shell with 2.1/2.2/2.3(eg, Cryogenmod) and the kernel is the underlying framework that controls how the phone is run. If you get the right programs and some common knowledge, you can swap between ROMs and kernels with relative ease giving you amazing control over your device.
So that's it in a nutshell, it basically gives you so much more control over the device. The whole... I guess rationale behind doing all of this is that generally (and pretty much everyone who does this can back me up on this one) the ROMs and kernels created by people (a TON over at XDA) are amazingly better than what the companies are giving us. There are major ones like Cryogen Mod that gives 2.3 functionality and a TON of features to tens of phones.... it really opens things up.
If you're interested or have any more questions or are looking for some guidance, drop me a PM, I'd be happy to help
I've been using ROMs and everything for about 7 months now so I can help out if you need it.