Summary:
The reversibility/irreversibility of evolution has been difficult to measure empirically. The researchers found a protein and its ancestral protein. They found key mutations that act like evolutionary ratchets. If the key mutations are left in place and the protein reversed to its ancestral form it became non-functioning (and would probably be a harmful mutation). Furthermore changes in the key mutation did not lead to any advantages in the ancestral protein. This means that there is a mechanism where a protein can mutate but is unlikely to mutate back, and this is a specific example of how evolution can be irreversible.
Exactly, it is a version of "use it or lose it". Think of cave-dwelling fish who no longer use their eyes. They lose the capacity for sight. And once lost, it isn't coming back.
To give more detail, they were looking at an important protein which appears in many species. Since they know the protein for many modern species, and they know when different species branched off, they can reconstruct the likely form of the protein at each branching. Given modern computer modeling, they can figure out what each version of the protein will do. They demonstrated that it used to respond to 2 hormones. 40 million years later it responded to only 1 of those. They identified the responsible mutation and found that random mutations that happened after the first would cause it to respond to nothing if you undid the mutation.
In short, a significant mutation opens the door for further mutations that make the first irreversible.
This is a good paper with an empirical example of a molecule level ratchet effect, but the message here is way overstated. It is definitely NOT the case that evolution "can't go in reverse" as a categorical statement.
1) First and most importantly, if selection pressure was reversed in sexually reproducing species, it is (a) unlikely that the ancestral allele will ever completely die out and (b) also unlikely that a sequence of recombinational events could not reconstruct the ancestral allele.
So reversed selection pressure would lead to an increase in frequency of the ancestral allele. Doesn't require the (much higher) level of difficulty of pushing six consecutive mutations to fixation in a single protein.
In bacteria or archaea it would be even easier -- just pick up the ancestral allele through horizontal gene transfer.
2) Second, even discounting sexual reproduction for now, given strong enough reversed selection pressure, either the species would go extinct or (more likely) some other protein would have been co-opted into pursuing the new function. That is, this road might be blocked but another one will almost certainly be open.
3) Third, this particular example has an accumulation of several mutations that act as blockades in fitness space.
But this is far from generally proven. You'd need to do some serious correlation of activity with multiple sequence/structure alignments across large families of proteins to show this ratchet effect.
4) Finally, the whole concept of QSAR (quantitative structure activity relationships) is based on the idea that activity-structure relationships tend to be more continuous than not, in the sense that small changes usually have small effects on activity [though in dealing with adding atoms to molecules, you're talking about irreducibly large step sizes in different dimensions]).
And continuous alterations in activity usually imply reversibility as a function of many small steps.
Since most mutations are no-ops, and behavior is frequently expressed through combinations, I'd have expected what they found. For every useful mutation, there were usually other random mutations that prevent simply flipping back the useful one.
Science isn't about collecting reasonable sounding ideas. It is about collecting useful ways of understanding the world that have been stress-tested until we can have great confidence in them.
The paper is the first time anyone has been able to provide solid evidence about whether the evolutionary mechanism you outline really happens. It's the difference between a note in Fermat's margin saying that he has a proof but the margin is too small, and Andrew Wiles writing up a several hundred page proof. Now that it has been demonstrated how to show which mutations are irreversible, the door has been opened to using molecular genetics to provide absolutely definitive answers to questions about the evolutionary tree that we don't have good fossil evidence for.
Furthermore the ability to test specific questions about the functionality of putative reconstructions of particular prehistoric genes does a lot to improve our confidence that we're reconstructing them correctly. We've now been able to compare versions of a gene from 440 million years ago and 400 million years ago and gotten specific enough answers back that suggest we've got both versions of the gene right. Monkeys and apes split perhaps 30 million years ago. How accurately can we figure out the genetics of that last common ancestor?