I know that DDR3 RAM is faster, but I don't know what situations have performance heavily influenced by RAM speed. When is this useful?
It is unlikely that in a normal modern computer that RAM speed is limiting anything. 3d games are going to be graphics card limited. General usage is likely to be network or harddisk limited.
RAM will make some difference to things like file compression or audio/video processing, but again, these are just as likely to be disk limited, so the RAM will only make a very small difference.
There is more information in the answers to this http://superuser.com/questions/16757/is-ddr3-worth-it question.
Your processor wants to process something? Sure, let's grab what's neccesary from RAM.
Yeah, that took a while. Faster RAM means less time waiting, and if your CPU isn't waiting, it can be doing something. An awful lot of stuff will be slightly faster, but overall the change isn't on the scale of, say, HDD to SSD sort of improvements.