Actually, a certain level of anxiousness or nervous energy is necessary in order for one to perform their best. So the solution is partly to realize that you can play just fine with the nervous energy, and begin getting used to it.
Since this anxiousness or nervous energy never goes away entirely, another part of the solution includes important ingredients for handling it: preparation and experience. As your technique improves so will your results and, therefore, so will your confidence. You'll commonly hear this referred to as "trusting" your swing or stroke. So, how to conclude the anxiousness on greens?
Firstly, you must understand your swing and stroke well. Knowing whether you are actually doing the things you understand. Ingraining good habits - get the solid fundamentals and movements so well-rehearsed that they become automatic or subconscious
Being well-prepared and confident will help decrease uncertainty and anxiety, which will reduce tension and help you relax and play shots. Technique development is a long term process. That is why golf requires practice and that is why it doesn't happen overnight, or by discovering some "new secret." There is no shortcut or secret. You have to go through the process.
In addition, play so many times that you have both performed well and messed up a number of times -- particularly in situations that you have difficulty with (like playing in tournaments, or even just playing with people you don't know, etc.). You may mess up before you perform well. Learn from your mistakes.
Also, learn to experience. You will never know how to get rid of it if you do not experience what it is. Only by repeated exposure to what causes you difficulty do you gain experience and expand your comfort zone; just as only by exercising can your body adapt and become more fit. Experience is a great teacher that will also help you to know your own game and become a better decision maker on the course.
Finally, do not forget to take a deep breath each time before you hit the ball out! This can really assist you in some degrees to reduce your tense. If you have a regular golf partner, ask him to help you release the nervousness. At the meantime, you can also help you in the same way!