Approaching girls is not like riding a bike, where once you know how, it's easy from then on.
think of it more like conditioning, like being able to run a mile in under six minutes or something. I takes a while to be able to do it,and it takes consistent effort to keep doing it.
Start small, slow, and work your way up. Don't expect to ditch your anxiety by reading a post.
Eye contact
Eye contact and smile
Eye contact and approach and say hi
Eye contact and approach and short conversation
Eye contact and approach and short conversation and number close, etc.
See it like you would any other training regimen.
Or do as I did. Grow impatient and increasingly frustrated at yourself to the point you just go in and wing it. Of course with the full expectation of failing horribly.
I had this idea that if I just throw myself into the situations the sheer pressure alone would conjure up at least something to say. So it did. By the time I had forced myself there it would be more humiliating to walk away than to just try something, and almost magically my brain found a way to decrease the massive tension I felt.
I remember being at a club with my female cousin a few years back when I was really socially anxious around strangers. I had one beer and was practically sober when she jokingly told me to approach some random girl at another table. I just went for it, had a nice little chat, very soft rejection and came back to our table. My cousin's jaw dropped, she did not expect that I'd actually do it. She asked how many beers I had (I arrived a bit earlier than her), I said I only had one while waiting for her. I just laughed about it.
That wasn't really a big deal. I didn't score, but nothing really bad happened either. I had no clue what to say when I went up to that table, but when I was standing there I had no choice but to come up with something. I couldn't take the pressure of standing there like an idiot or walk away again, so I started talking.
Just notice when someone is talking almost non-stop, chances are they're trying to compensate their anxiety by filling silence with talking. When you're in a situation like that it's hard to shut up, your anxiety will actually help you avoid being at a loss for things to talk about.
So how to approach a woman? I'd say just do it and wing it. Exposure therapy has proved to be very effective, I say that from a personal perspective also.