Well, in my opinion, it is kind of reaction which often happens to many people. When you are speaking in the public, you will feel nervous. So that it can irritate your eyes' nerve system. In other words, there is an eye irritant like a chemical. And it forces the eyes to water in attempt to relieve your situation. Generally speaking, it is related to the nerve system and the tear system. In some degree, we can say this is a very common thing which you should not worry about. And tears are a defense mechanism your body uses to protect the eye and wash away the enemy attacker. To overcome it, you can just try some ways to make yourself not so nervous. For example, you can have some deep breath, before you get on the stage.
Well, you should know that sometimes our body will give some special reactions to some unusual occasions which make you feel nervous. It is not uncommon for your eyes to water in front of a large audience, since you are trembling and could not resist being afraid of making mistakes. So, there are no immediate solutions to that, the only way for you is to train yourself to be an experienced public speaker.