Licenses like the GPL are often described as "copyleft," implying almost the opposite of copyright. Free Software often seeks to re-grant the rights that copyright law takes away, but it’s still based firmly in this legal framework.
Copyright law, irrespective of its original aims, is about providing creators with choices in how they distribute their work. Free Software is one of these possible choices. It’s no more anti-copyright than a proprietary EULA is, and in some ways is closer to the original spirit of the laws.
Copyleft is often detracted as taking away choice, but this isn’t really the case. A developer has the choice of using a copyleft piece of code and abiding by the condition, or writing it again from scratch. Under copyright law without the copyleft license, the only choice for the developer would be to rewrite the code from scratch.
The hacker mentality takes great pride in using tools in a way other than the creator had in mind, and Free Software is the outcome when this principle is applied to copyright law. It is no more anti-copyright than a complicated awk script is anti-UNIX.