The roots of the anxiety are myriad:

Maybe you've never really quit a job before. You always had a built-in out. "Well, school starts again and I'm heading back to college."

Maybe the company just hired you a few months ago and you feel kind of bad about making them go through the hiring and training process all over again.

Maybe it's a small company, you've been there a long time, are close to your boss and co-workers, and feel like you're leaving them in the lurch.

Maybe your boss is a tyrannical hothead (that's why you're quitting!), and you wonder how he's going to react when you tell him.

Any way you slice it, when you're quitting a job, you're sort of firing your employer. It's somewhat comparable to breaking up with someone. And just like with breaking up with a girlfriend, there's a right and wrong way to do it.