If a person does not abide by the terms of the GPL, however, then the permission that the GPL grants does not apply to that person, and so could be found guilty of copyright infringement when making copies of the work that do not fall within the boundaries of fair use.
Copyright is about more than just fair use, you know.
It may help to think of copyright by means of a Venn diagram. It's like a subset that is, itself, full of sub-subsets that compromise holes in what the subset contains.
The largest set is the public domain. Anything in the public domain can be used freely, in any way, so far as copyright is concerned. (Because we're talking about copyright, and not, say, personal property, note that a copy of a work can still be owned and off-limits even though the creative work embodied in that copy is not protected. For example, the Mona Lisa is in the public domain, but the actual wooden painting made by Da Vinci is heavily guarded. Copyright might not give anyone the right to stop you from making a copy of the Mona Lisa, but the Louvre doesn't have to let you take a photograph of it.)
Works that are copyrighted fall partially into the subset of copyright. Partially, not wholly, because copyright only applies to certain types of action with regard to a work. For example, making a new copy of a work by reproducing it is one of the exclusive rights of copyright. But using a book (which is a copy of a work) to prop up an uneven bedframe is not an infringement of copyright, because the law does not grant copyright holders an exclusive right of 'propping-up.' Likewise, some kinds of works are not copyrightable. In the US, from 1790 on, only books and maps could be copyrighted. Only when the law was amended in 1802 were some engravings copyrightable. In 1831, all engravings, and also musical compositions; 1856, dramatic works; 1865, photographs; 1870, paintings, sculptures, drawings, and models and designs of works of fine art; 1912, motion pictures qua motion pictures (previously they were claimed as collections of photographs); 1971, sound recordings; 1976, pantomime and choreography as themselves, as opposed to being dramatic works; 1990, architecture. Anything not on the list of protected types of works is in the public domain. (Software, in case you're wondering, is treated as a literary work, like a book, but doesn't yet have its own category) There are a few other limits on what is copyrightable, but they're not terribly important for our purposes.
Of the list of enumerated rights which compromise copyright, and which is short in comparison to the list of all rights concerning those works and their copies, as applied to those works which are eligible for copyright, which is less than the totality of all works, there are still some subsets which further limit copyright protection.
Fair use is one of them -- any otherwise infringing use, which is fair, is not infringing. That's a fairly good-sized hole in the already swiss-cheesed set of copyright. First sale is another big one -- the right of the copyright holder to control the distribution of copies is almost entirely obliterated once the copyright holder has sold the copies in question. There are some exceptions to that exception, and some exceptions to those exceptions to the exception, but it's still shrinking copyright. Another one you might enjoy for software is that if a person owns a copy of a program, he can copy it and modify it in order to make it work, without needing permission. Likewise, he can make backups without permission. So really, so far as copyright goes, people in the US only need to agree to the GPL to copy a work if they're going beyond the statutory exception that keeps copyright from stopping them making certain copies.
There are exceptions like this throughout the Copyright Act. They can be broad, they can be narrow, they can be subject to various conditions and exceptions themselves. But the point remains that copyright is an all-encompassing, all-covering blanket. It's a like a piece of swiss cheese, full of holes.
Aereo is going through one of these holes -- a fairly big one, since copyright doesn't even apply to private performances and no more specific exception is needed; it's like they were in the propping-up business I mentioned earlier. It doesn't imperil the viability of the GPL. If the next version of the GPL tries to assert that people must agree to it in order to do things which copyright doesn't prohibit, then the general thrust of Aereo that everything which is not forbidden is allowed (one of the foundations of our entire system of law, btw) might matter. But I don't think the GPL is likely to go so badly wrong anytime soon.
Perhaps instead of strongly asserting a position ("it's copyright infringement, plain and simple" ... " it could effectively spell the end of open source works" ... "I sincerely hope the supreme court slaps this one down.") which doesn't seem to have been based on a lot of analysis ("I don't know how unlikely what I fear if this is upheld might actually happen is, to be honest... but it's the first thing that *I* thought of"), you should refrain from staking out a position until you've really looked into it.
Hell, you could even try asking RMS, who is pretty knowledgable about the GPL, as to whether a win by Aereo would imperil FLOSS (or whatever people are calling it now, it's all People's Front of Judea to me).