Your OP doesn't seem to match your title - your title refers to you having problems with not keeping friends and falling out with people, whereas your post seems to suggest that you struggle to make friends in the first place. Which is it? Or is it both?
I used to be very sensitive and fall out with people a lot. I think my expectations were way way way too high and I took things very personally. I made a conscious decision about 10 years ago to stop doing that and to cultivate more friendships so that I was less reliant on a small group and less sensitive when I was perhaps not invited to a particular event or whatever.
The first thing I actively and consciously decided to do was to stop saying negative things about others and to try to find positive things to say wherever possible. Moaning or complaining or bitching are all very wearing to listen to and don't cause people to warm to you. People like talking about themselves, so try asking about them or complimenting them if you like their hair/necklace/top etc. You don't have to be a bum-licker - don't pay an insincere compliment, but try and think of nice things to say, smile a lot, try not to give one word answers if someone talks to you. When someone approaches you (e.g. at work) try and smile and ask how they are etc - make the interaction a bit more social.
It's not realistic to think that you can be everybody's friend, but you can be popular and generally well-liked. That doesn't mean you'll have everyone's mobile number or go out drinking with every person every weekend - but it is possible to get along well enough with most people that you'd feel comfortable sitting next to them at the Christmas party and possibly there might be a couple of people that you would socialise with outside work. This depends of course on how big a place you work in - if it's just you and 2 other people and you are all very different, it might not be fertile ground for meeting close friends!
You should also remember that most people are more interested in themselves, their own lives and their own issues than making detailed judgments about you. I read (probably in a trashy magazine...!) once that self-consciousness is actually a form of arrogance, because you think others are far more interested in you than they really are. Try to focus more on finding out about people and deciding whether you like them and want to get to know them better. Try to see the best in people. Make the first move to suggest going for a drink after work or going for lunch (this is a good one with work colleagues - everyone has to eat lunch, after all).
It does get easier. The first year or two of my 'personality change' was hard, but now it all feels normal.