Other pages (such as UseRealNamesDiscussion, UseRealNamesRefactored) usually concentrate on the advantages for the community (or at least the regulars that stand for the community consensus), when users use a RealName.
Note: this page doesn't say who is right or wrong. It reacts to a common situation. It doesn't say that every community prefers real names (in fact many online communities don't care, or actively encourage IdentityGames) or that a user must or should use his RealName (he is autonomous to decide anyway).
Simplicity. It's the simplest thing. You need a name, you use your name.
Trust. If a person doesn't use their RealName, there is a reason for it. There are many possible reasons, most of them mean problems. So the community will not trust people without RealName - except if there is a really credible explanation. Community and individual usually don't agree on what that means, so things get complicated and tiresome. Communities dislike to be burdened with tiresome problems. Using your real name avoids having to repeat complicated explanations over and over.
History of cooperation. With a RealName your contributions are recognized easily from the signature and RecentChanges entry. Other people recognize and remember you. Any positive idea or communication will help you in the future. In anything you do, any question you post, any request for cooperation, you depend on how other people feel about you. Only if they remember you positively, you'll get reactions above average. (See IteratedPrisonersDilemma )
Copyright. The issue of copyright in online communities is totally exaggerated. It is very rare that contributions create a financial value. But at least in theory you are the master of your words and you may decide who may use them. Posting your contributions as PublicDomain or OpenContent may help you more in the community but you needn't feel obliged to.
Authorship. Being recognized (and honored) as the author.
Reputation. Using a RealName is the most credible way to build a combined online and RealLife identity. Over time you will be part of many communities and leave your footprints behind. If people can understand your work and intentions, if they can look into your past, they'll understand you better and trust you more. With your positive contributions to the overall community you build a reputation.