There's an old Polish saying: "How they see you is how they perceive you." Although our art is largely one for the ears, the stage presence of a performer can make a tremendous difference in the audience's experience (as well as their perception of the success of the performance).
Simply put, your performance extends far beyond your playing the notes. The energy (and posture) with which you walk on stage, your eye contact with the audience, your smile, your bow, your movements during your performance, and the way you finish, stand up, smile, bow, and walk off the "stage" area all are all part of the package.
If you combine all these elements, they add authenticity and professionalism to your final product... Forget about them and any one of them can take away from it.
We are actors and storytellers of the music. The more we can portray and convey the adjectives we would use to describe the music, the more involved the audience feels. Try video-taping your practice/performance and check to see if your physical choreography, posture, and facial expressions really match the mood of the music.
Being comfortable with your stage presence takes practice, but yields great results in connecting with the audience. In addition, YOU will feel more comfortable on stage.
Realize that performing is a gift you are giving to the audience. They eagerly anticipate getting to hear you. Give it to them with joy!
Here are a few helpful articles on stage presence:
http://www.hilaryabigana.com/apps/blog/ (a LOT of good advice, including videos)
Elizabeth Borowsky is a pianist, teacher, and composer. She is a Nationally Certified Teacher of Music in Piano (Music Teachers National Association).