Andy's answer covers benefits to individuals quite well; I'll try to cover some benefits from the community perspective.
More members, more posts
Signing up is easier if you don't need to provide a bunch of information upfront, and people are more likely to sign up because they will not be concerned about the privacy of their personal information. (Once the community means more to them, or they trust it more, they may add more personal information anyhow.)
Anonymity also removes many of the reasons people might have to avoid making a post. This means more posts, which can mean a more active and faster-growing community. It can also mean more bad-quality posts and more spam, however with the right measures that can often be made manageable. Just remember that not ALL posts made anonymously are bad, some can be excellent and may not have appeared if the author was not able to post anonymously.
Lots of human suffering is caused by "taboo" subjects that nobody is willing to speak up about. Think depression, homosexual attraction, domestic violence. The secrecy causes much shame and isolation, and reduces the chances of people getting help. Anonymity can help such topics see the light of day, and in so doing benefit many members of the community, some of whom may not be willing to post even anonymously on the topic, but will read and benefit from the discussions. This will increase their desire to keep returning and contributing to the community in other ways.
Of course, this is not restricted to the types of examples I suggested—they were quite big issues which made my point. The same thing happens at smaller scales with smaller issues, often tied to the topics the community is based around, and thus very relevant to the community.
In addition, there are other types of secrets, "tricks of the trade", or things that everyone is thinking but nobody wants to say, that would benefit everyone if they saw the light of day.
In a community where people know nothing about you except what you post, there is much incentive to contribute high-quality content. Your appearance, age, location, and "friends" aren't there to make you look good (or bad). Essentially, people who choose to post anonymously must work a little harder to develop a good reputation, or for their contributions to be accepted. In turn, the community at large gets in the habit of evaluating contributions based on their merit.
Of course this doesn't "automatically" happen; it needs a healthy community that is actually interested in high-quality content. But it can play a part.
That's really the bottom line; you don't just push a button and add anonymity to a community without making any other changes. Anonymity has benefits and drawbacks, but if you're considering anonymity you must also consider other changes that will make anonymity work for your community. You must have a plan for how it will fit in your community.