It's a violation of Newton's 3rd law of motion? This is a fairly complete answer to the question in the absence of any deeper argument by the proponents of the device, but I'll get to those in a minute. Firstly, it's important to recognize that our universe follows a set of predictable and consistent laws. This might have been controversial in the 1700s. One of those laws is that forces must be accompanied by an opposite and equal force an another object. Now there are way that modern physicists can revise this in creative ways. For instance, there could be many bodies acted upon by a force that can't be divided up into 1-to-1 interactions, but these are all objections that don't change the bottom line. You can't push against nothing.
Another revision might be massless particles. Radio waves are photons, and they have momentum like photons. You can propel yourself by emitting any kind of light - but the forces are miniscule compared to what is being claimed here. It also requires tremendous amounts of energy.
So let's move on to the "modern physics" adjustments to Newton's 3rd law, and possible violations of it. It is sometimes claimed that general relativity doesn't preserve Newton's 3rd law. Maybe this is semantics. Practically, it's obvious that gravitational waves are scientifically verified and predicted by general relativity. Much like photons, these can carry momentum along with them. Since this is a "ripple in spacetime" and not momentum carried by some matter, there's certainly room for argument. However, momentum is still conserved. Also like photons, this is mostly useless in terms of the actual thrust involved and the difficulty of building a practical machine.
Instead, most of what I'm reading about this subject makes reference to "new" physics, mostly quantum gravity type stuff. There are many theories that would predict that some sort of reactionless drive is possible. However, proving such a principle through a reactionless drive prototype is contextually bizarre. If you could show even the slightest variation from classical physical theory, then that's interesting by its own merit. What I'm saying is that any such machine should first be interesting to physicists, then be interesting to spaceflight. It's entirely possible that we will discover some novel way to break Newton's 3rd law, but then we'll have no practical way to use it in spaceflight. After all, this was the exact case with my above 2 historical examples.
More than likely, they're not on to anything. Dr. Harold White uses real physics and makes claims backed up by genuine science, in addition to having access to a large body of peer scientists to back things up. However, his statements are difficult to parse correctly, and this combines with irresponsible reporting to produce headlines which are often blatant falsities. In his paper abstract, he refers to force produced by virtual particles. Great, that's fine. This is not tremendously surprising, because this is within established physics. It might just be that they achieve "propulsive momentum transfer via the quantum vacuum virtual plasma" without diverging from current theory. But that doesn't make it a reactionless drive! Current theory still preserves momentum. Maybe no one has done a detailed study to see how this particular case balances momentum, but we have every reason to believe that it does. Even if they are onto something, that lends no credibility to the bogus experiments that are searching for some true reactionless drive.