The above is all wise stuff: I am an actor and know what stage fright feels like. Goldtopia has put what everyone is talking about into a nutshell. In my experience it never goes away permanently, but there are lots of ways to turn it to positive effect and even maybe beginning to learn to recognise the physical symptoms as pleasurable, and not painful. Many of the symptoms of fear are also recogniseable in sexual excitement, after all.
It is not exactly fear of being heard, it is the adrenalin rush caused by the sensation that our concentration has wandered away from what we are doing, and into what other people think of it We are diverting brain power into telling ourselves stories about what other people are thinking, instead of how WE assess how WE feel the MUSIC is going. This makes it much harder to play accurately because, however well you think you have learnt it, good playing requires concentration. This all happens at lightning speed, too fast for our brains to understand it at the time, which is why when we think back to it, we recognise the physical symptoms and call it fear.
We rationalise it as fear of humiliation, and, so that we can get our heads round it, build a picture of what we are afraid of, usually missing notes. And, because we are thinking of wrong notes, we play them. Or, because we are thinking 'blimey that went better than usual' we immediately spog up because the fingers are no longer getting accurate direction.
The key is that it is distraction from the act of playing by the novelty of having other people there. You may find that when you are practising, you tend to play more quietly, or pull your expressive punches if the family are in, or reserve the difficult pieces for when you have the house to yourself. Same thing...
There are two vitally necessary parts to the remedy.
One is, as said above, in practice PLOUGH ON. It's amazing how often a fluff becomes an ornament without trying, or the adjacent chord sounds (nearly) as good - actually you can make great discoveries that way. Of course you have to go back and grind the technically difficult passages into submission, but as a very good teacher put it to me recently, the worst habit in the world that you can teach yourself is to slow down or stop when you get to bits that have been a problem. Often they are the points where the music needs the greatest flow for effect so, she said, play the piece all the way through at the speed at which you can play the hard bits (very slow in my case) until the hard bits have become muscle memory. Then speeding up is just a matter of exercise. Pound out of yourself the habit of audibly or visibly reprimanding yourself for an error. Every ounce of mental energy you expend on that makes more errors inevitable. Not likely, inevitable. Major, successful actors, mashing their lines up (and they do) plough on. It's not because they don't care about making an error, it's because they care passionately about not making another one if they can possibly help it.
Second is to play with other people so often that it ceases to be a novelty. Practice when the family are in. Go to ensemble sessions (actually concentrating on making what you are playing meld in with other musicians counters 'stage fright' beautifully, because you are concentrating on the music, not making up mental pictures of other peoples' reactions). Moving on to paying solo is still a step, but if you expect to be petrified it will feel better than you expect. And it will force you to play at other people's tempi, which makes it easier not to berate yourself for every error. You will learn the self-preservation technique of blotting the audience out. Think always 'what is the music doing' (which we can do something about) and not 'do they like it?' which is completely and permanently out of our control. If they don't like it, that is their decision. One might try giving more reign to expression if you feel yourself start to sweat, or forgetting to breathe. Concentrate on what you can effect, and it will leave you no time or brain-space to bother with what we can't. One can get the most wonderful feeling of your fingers being slightly behind your brain suddenly, rather than the other way round. Some feel it as 'having got one over' on the listener: I know one player who calls this pulling the "soddem" lever, which is not an expression I like, but it gets him through.
We have, I suppose to live with it if we are going to perform at all, and to get our thrill from the music first, and leave the listener to get their thrill where they decide. I noticed at a recent gig by an internationally famous musician that one song he had written himself years ago, and which was very important to him had a very unusually wandery middle section, and his warm-up patter for the next number was accompanied by some very elaborate re-tuning. I began to wonder, and asked a mate who had played the same bill as him. 'Oh yes' came the reply, 'absolute martyr to nerves, though you'd never know it before the gig these days. He was probably just noodling around until he got his concentration back. He's much better than he was'. Only the performers in the audience had an inkling.
By the way, I have not found drink in any quantity constructive. It becomes a learned behaviour, one's physical tolerance to its effect increases over time, and from the moment that the first milligram is absorbed into your blood stream in blunts concentration, creating a further lurking doubt of being out of control, creating the symptoms of fear.....