Don't think in terms of getting rid of it. You might never COMPLETELY get rid of this fear. But you need to chip away at it by approaching. Then it will weaken big time.
Also understand that every night you go, every party you go to, etc., you have to WARM UP. Even experienced guys do this. You might start out the night scared sh*tless. Start talking to people. Girls, guys, young, old, etc. It doesn't matter. Then move on to more attractive females. After talking with several females, you get in "the zone" and it becomes really fun. I know that is hard to believe when you are on the other side of the "wall," but it's true. Then the next night you might feel that you are "starting from zero" and have to warm up AGAIN. That's normal.
Also understand that "non-naturals" are actually capable of a HIGHER tolerance for rejection than naturals IF they apply themselves, do the work, and develop that thick skin. The "natural" is the guy who always got girls. They responded well to him since he was very young. He doesn't really have to try much. As a result, he's sort of spoiled. But a guy who struggled with women from the beginning can develop a thicker skin than the natural if he applies himself.
Anyway, that's too much theory. Let's get back to the practical.
Steps
1. OBSERVE other guys meeting girls. Go to a party or somewhere you think other guys will be meeting and picking up girls. Just hang out and observe. This should do two things for you. First, you'll see that what they are doing isn't rocket science, and it's no big deal when they are rejected. Plus, it might trigger a competitive instinct. "If these guys can do this, I certainly can too."
2. Focus more on getting warmed up as mentioned above.
3. Finally, you gotta believe that you are a cool and attractive guy. Even if you don't believe this at first, pretend it's true. Fake it until you make it.
4. Just talk to girls.