The brain is a physical organ. As such, it follows the laws of biology, chemistry, physics, quantum physics, and of any other hard science we have yet to discover. The brain cannot make choices. It is not free. So when someone says that mind is nothing but brain, he is saying that the things we associate with mind—choosing, preferring, thinking—aren’t real.
In philosophy this is called epiphenomenalism.
Firstly: No. 'Epiphenomenalism' is not the view that "mind is nothing but the brain". That's physicalism. Epiphenomenalism is a kind of "one-way" dualism: the brain gives rise to the distinct phenomenon of conscious experience (or 'qualia'), but these qualia do not affect the physical world. Unlike interactionist (or "two-way") dualism, the view is thus consistent with the causal closure of the physical -- the principle that physical effects must have merely physical causes. For example, my writing about the experience of seeing red is caused, not by the experience itself, but rather by the underlying brain state.
(People often claim that this is counterintuitive. I'm not so sure. We may think of mental states as having both physical and experiential components: their physical effects are due entirely to the physical aspects of our thoughts. The non-physical (experiential) component, on the other hand, constitutes what it feels like to be in that state. There's then an obvious sense in which our mental states have causal effects, insofar as their physical aspects do. That doesn't require that the causal 'oomph' come from the experiential aspect -- indeed, how could it? Experiential feels aren't the kinds of things that push atoms around. You need other particles to accomplish that!)
More importantly: there's no reason to think that physicalism is incompatible with "making choices". The reference to physical "laws" suggests that the author may be specifically worried about causal determinism (which is orthogonal to the physicalism/dualism debate), but even then, note:
(1) Whether or not determinism is true, there is an obvious sense in which the brains of humans (and, in an attenuated sense, even many non-human animals) implement "choices", understood as a certain kind of information processing (involving decoupled representations) leading to goal-directed behaviour. Abstracting away from the experiential feel of it all, this cognitive-behavioural phenomenon is perfectly physical -- biologists and cognitive scientists study it!
(2) Perhaps the author is instead concerned about whether our choices are "free" in the strong sense necessary for moral responsibility. That's a more respectable worry, though it's worth noting (as the author seems unaware) it's also quite possible to view determinism and moral responsibility as compatible. After all, there's clearly some significant difference between the psychological processes underlying ordinary behaviour and compulsive behaviour, for example, and we can still track that difference (and accord it moral significance) even if determinism is true.
(3) Even if that is the worry, the author is way too hasty in leaping to political conclusions. He writes:
If mind is brain, there is no “psychological” freedom or responsibility—no humanity. And if those don’t exist, there can be no political freedom or self-responsibility. What does not exist cannot be violated. [...] The hard sciences are great human achievements, but for the sake of liberty, they must not be permitted to overstep their bounds.
But political liberty is an entirely different matter from psychological (let alone "metaphysical") freedom. The former is threatened by external coercion, whereas the latter is threatened by (e.g.) internal compulsion. These are different! Even if it turns out that moral responsibility is an illusion, that just means that people aren't to be held morally accountable for the things they choose. It's not a reason to impede those choices (given the usual proviso that they don't harm others).
There are plenty of (e.g. utilitarian) arguments for classical liberalism which clearly don't depend at all on assumptions about moral responsibility. And even some that might initially seem related actually aren't. Libertarian talk of "individual responsibility", for example, presumably concerns a particular kind of behaviour, and people can still behave more or less "responsibly" in this sense, even if there turns out to be no such thing as "moral responsibility" in the sense of moral desert.
In fact, the only clearly relevant libertarian argument that springs to mind is the claim (found only in the more conservative corners of the broad tent) that we shouldn't redistribute wealth because poor people "deserve" their lot in life. If nobody really deserves anything, then that argument is pretty swiftly undermined. But there is (thankfully) more to "liberty" than just the liberty to ignore those in need. Can readers think of any other (less morally repugnant) libertarian arguments that depend upon the existence of strong moral responsibility?