While I still feel this exact kind of "making the approach" inhibition from time to time, I think the way I've gotten over it is to build up a history of cases where I just sucked it up and did it and was able to look back and realize that hey, it wasn't so bad! Oh, going with a friend really really really helps too! One of you makes the approach but just having the friendly face to fall back will enable a soft landing.

That being said, starting out in less challenging settings is definitely the way to go (ex: if you don't like loud bars, don't practice getting better at approaching at loud bars! Try programming meetups or interest meetups where the ambiance is more natural to you are likely to have things in common with the people there).

The emotional side of the equation is to have a history of success that you can draw on for courage. The rational side is to know and repeat to yourself that it's a numbers game just like dating. At "networking" type of settings, I personally expect to meet one interesting person with whom I'll have a sustained relationship with every 2 hours. There will be many friendly people along the way where we will mutually forget each other after 2 days, and that's just the way it is.