The biggest part is what to say in the apology. I recommend you to write down the apology before setting up the meeting. Convince yourself that you truly are sorry. If you are a naturally shy person, you want to feel comfortable with what you say and reduce the chances of making blunders.
There are usually three parts of an apology: the expression of guilt, the acceptance of responsibility and the righting of a wrong. The biggest part is admitting that you were wrong. Next, accept responsibility for your actions, which leads to the last step, correcting your mistake.
Edit your apology after you write it out in full. Do not pretend like you are writing an essay for school because then you will be dishonest. It is important that you appear sincere without sounding generic.