a place for zinesters - writers and readers
Anti-copyright means you are opposed to copyrighting creative material and believe it should be made freely available to everyone. Much of the current anti-copyright movement grew out of software development -- the idea that Microsoft or some other corporation shouldn't have exclusive rights to control and make profits off of things that all people need and thousands (if not millions) of people have contributed to developing. In a broader sense, it is opposition to the very idea of "intellectual property," especially in the hands of corporations.
I think with zines it come down to the idea that people are free to make copies or reuse in whole or in part zines you have produced.
One of the difficulties that comes from living under the current copyright laws is that if you put something in the public domain, someone else can pick it up, modify it slightly, then copyright it and take exclusive control. Some alternative means have been devised to prevent this -- particularly Copyleft and Creative Commons, which attempt to use parts of existing laws to ensure that items intended for free public use remain that way.
Here are some links for further reading on the subject:
Anti-copyright page on Wikipedia
What is Copyleft?
Is Creative Commons anti-copyright?