Personally, I think of it as giving your employer "first dibs."
"Hey, I've decided that I'm now worth $X. Since you've already hired me and it's less hassle to stay than to change jobs, I'm giving you the opportunity to meet my price before I look elsewhere."
Assuming that you didn't just pull $X out of your ass, the company can then make its decision accordingly. But if you really think you're worth $X, I find it strange that you would get declined by the company and then say, "...oh. Well, I guess I'm not worth $X, then. Never mind." At the very least, you'd be looking at other companies that are willing to meet your price. After all, you got your number from somewhere, right?
"Hey, I've decided that I'm now worth $X. Since you've already hired me and it's less hassle to stay than to change jobs, I'm giving you the opportunity to meet my price before I look elsewhere."
Assuming that you didn't just pull $X out of your ass, the company can then make its decision accordingly. But if you really think you're worth $X, I find it strange that you would get declined by the company and then say, "...oh. Well, I guess I'm not worth $X, then. Never mind." At the very least, you'd be looking at other companies that are willing to meet your price. After all, you got your number from somewhere, right?