rooting an Android phone is the act of attaining the superuser/root privileges that are locked by the manufacturer. rooting allows support for Android theming, full control of the kernel, full control over the applications, access to more UNIX commands via adb, etc.
if you're skeptical about rooting your phone, remember that rooting is totally normal and should have been done from the beginning by manufacturers. imagine buying a computer with Linux and not being able to change the OS, nor even use sudo. yeah, that's dumb. well that's the same thing with phones. rooting your phone simply gives you the same amount of privileges as your computer. as long as you don't give suspicious apps rooting privileges, you are going to be ok.