Freedom of speech
Anonymity goes hand in hand with freedom of speech. If you want to say something without it being tied to your identity then you can use anon status for that. It is also abused for bullying and it can be quite annoying but people would bully you too by making you throw away accounts.
It allows you to have privacy. Everybody can view this website, everybody. If you put a picture of yourself on your profile and you use a name which you have used on other sites people can find you and read your questions, which may contain sensitive information that could be useful to them. Just think about a relative finding you on here, or your boss. People get fired for what they post on social media so that anon button protects you more than you may know.
Substitute for pseudonyms
People that write books or articles on sensitive or controversial matters may want to use a pseudonym; this allows them to use another name (which is legal btw) to write and publish books and articles which would otherwise never be published if pseudonyms were illegal. The anonymity button allows you to use anon status without the need of a pseudonym, and that's very convenient. If anon status on GaG was not possible many myTakes would not be written and many questions would not be asked at all, because it would be a hassle to make an other account every time.
Tyranny of the majority
This concept is opposed to the concept of silent majority. A silent majority is where the majority in society remains silent on certain matters or issues, tyranny of the majority means that the majority attacks a minority due to controversy or sensitive matters, so the opposite of when they would stay silent. Anon status allows you to stay safe from tyranny of the majority when you talk about controversies.
Tyranny of the majority may lead to death threats, bomb threats, harassment, stalking, etc.
Anonymity is bashed on here a lot. I sometimes fear that people are forgetting that people may be watching over our shoulder. We often use the word stalker, but many people would never undertake steps to protect themselves, and their information from them.
But can you blame them? Many people grew up where posting your entire social life on the internet is the norm. They do not understand the value of the information they place on the internet.
What happened to me
I had Facebook but deleted it when information that I put on there any my family was used to break into our house. That was it, I deleted it because it added no real value to my life, and I could not live with the fact that my own information could be used against myself and my family.
If you don't want anons on your question you can always disable the button. And if you like making use of the anonymous option GaG provides then don't abuse it. All in all we need anonymity to protect our information because people can and will use it against us when we are not careful with our information.