So beautiful 

Check the charging current

It's basically having admin access to your phone. You can then modify stuff or uninstall/freeze system apps or install custom Roms (Operating system)...which you will NEVER be able to do without having your phone rooted.

Personally speaking, root your phone ONLY if you need such access. If you're fine with your phone the way it is now (unrooted), then no need to root lor.

Personal experience:

1) I was under a very limited 200MB data plan and thus, I had to skimp on web browsing. I started to notice there are tons of apps that are eating away data, which I did not authorize them to do so. I come to know of an app called LBE Security Master that can RESTRICT such permissions by only enabling them to access data through WIFI only. This requires root.

2) Unnecessary permissions by apps such as Phone ID, IMSI, IMEI, phone state and all those really suspicious stuff. How do you only allow certain apps to have certain permissions? Use LBE Security Master. This requires root.

3) I noticed my phone gets kinda hot sometimes, so one of the ways to reduce the temperature (albeit some performance reduction), is to underclock it. How? Install apps like Set CPU or CPU Master Free. This would also save you some battery too. This requires root.

4) Ram Usage- As SIII only has very little ram, around 800MB+, it is therefore essential to have good ram management. One of the ways is to freeze some system apps that eat up a lot of ram, for example those apps that some might not use, such as S Voice, S Suggest, Video Hub, Game Hub. I froze quite a lot of them and this helps in lesser ram Usage. I'm using System Tuner. This requires root.

Just my 2 cents on how I wanted to root my phone. Very practical reasons

Update: And oh, you need to root your phone in order to install Custom Roms to your phone. Basically it's like installing a different OS into your phone