How do moderators on forums find out if you create an alternate account?

The server can tell from what IP you came, what brower you are using and, if a cookie is set, whether you are returning to the site using the same computer. They can't tell you are the same physical person behind the keyboard, but they can certainly tell whether it's the same computer and the same user on that computer. If you log in to their service using different credentials each time, it will be flagged by the server, and the penalties you mention can be applied automatically or after review by a "real person". You might be able to avoid this by creating multiple login accounts on your computer and staggering the times when you use the service. That way different tracking cookies will be set for each user and it will seem like a public computer.