Just about every web site comment form I see (other than for web sites requiring registration to comment) asks for name [required], email [required, but will not be displayed] and web site [optional]. It doesn't matter what you put in the email field, so why ask? Just tradition?
Sometimes there is an option to be informed about replies and e-mail is a natural option for such situations. It may also come down to trust that people who leave comments provide correct addresses so that you know that somebody has already commented before. This way you do not need to rely only on IPs and cookies.
What is more, there are websites and services that let you create an account and claim comments left under the e-mail address you have registered with. Disqs does it, for example.
Maybe it will be needed for you to get contacted, maybe you will be sent an email when you get an answer to a question or sometimes you must be held responsible for what you've said if you break the rules of that forum/website. Or maybe a certain forum post can win a prize and how can you prove that it's yours if it weren't for your email address? :)
In addition to the other answers posted, a lot of sites use Gravatar to show an icon for the author alongside their comments. I believe Gravatars are based on the email address.
A lot of the time, it wants to know you're a real person--whether that is for the sake of the other two answers to this question by Michal Gancarski and Keiki. Usually, with some amount of bot/spider capturing software on the webserver, fake e-mails can get flagged, such as on domain hosts, that help add that little bit of security.
Of course, it's much more likely that they use it for other reasons as stated by the authors I mentioned. That is, to know who has commented before, to verify through e-mail sending that you're a real person (even more reliable than what I mentioned above, which might only weed out less than 1% of bots that scrawl your site), etc.
It's funny, that is even the case on Stack Overflow!