Sorry, forgive me but I've never been an Android person, although I starting to gain interest.
What is rooting and why do people want to do it on their Android phones?
Rooting provides you super user access on the phone. Just like an administrater has higher access on a windows OS and can perform restricted operations, rooting provides you that access on android phones allowing you to perform restricted operations.
One of the main reasons a very basic user roots his phone is to be able to remove bloatware (preinstalled applications that you dont want to use but are unable to install normally). Another reason can be to be able to use applications like Titanium backup which can be used to create backups of applications and data and to also freeze applications or services (to free RAM) which you dont want to un install (might want to keep them on the phone expecting using them some time later in the life of the phone). There are also several other apps that need root access.
A more advanced user may root to be able to install a custom recovery which can then be used to install custom kernels or ROMs, Mods or do a complete backup (create system image of all files and data including system files) of the phone so that if any thing ever goes wrong on the phone you can instantneously revert it to the last system back up.
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Thanks again. One more question - is there a way to completely go back to the stock software using this method?
I've checked and double-checked - model number is GT-I9300, so that seems to be supported, right?
Yes there is a way to completely go back to stock.
Yes GT I9300 is the international galaxy S3 version and it is supported. Also it should be easier to find unlock procedure for GT I9300 as the international version is really well supported by xda developers.