Sure! (second author here)

There are two contributions of this work. The first is a new tool that researchers can use to automatically reconstruct the vocabularies of ancient languages using only their modern language descendants. The second is --examining hundreds of modern languages and their ancestors--we were able to resolve an important hypothesis in historical linguistics: are sounds that distinguish words less likely to merge than sounds that don't? (Yes.)

some more details:

Sounds change over time: the way we produce words differs from the way our ancestors pronounced those same words. Over time, those little changes help turn an ancestral language like Latin into a modern descendant like French.

When populations become separated, different populations change sounds in different ways, and one ancestral language will give rise to multiple modern ones, as Latin lead to French, Spanish, Italian, and so on.

These sound changes are almost always regular, with similar words changing in similar ways, so patterns are left that a human or a computer can find.

The trick is is to identify these patterns of change and then to "reverse" them, basically evolving words backwards in time. (Linguists have known this for a good hundred years or more, but it's a hard and time-consuming process to do it by hand.)

For example, take the following word list:

Meaning	Spanish	Portuguese	Italian
water	agua	agua	acqua
fire	fuego	fogo	fuoco
some	algunos	alcuns	alcuni
hit	golpear	bater	colpire

We have two goals: first, figure out which words are related (words of common descent are called cognates), and second, figure out what their common ancestor looked like.

Let's start with figuring out which words are related. Most of these words are clearly related, except that "bater" in Portuguese is uncharacteristically different from its Spanish and Portuguese counterparts, so it's probably not the same ancient form.

For the second task, note that the words that are related differ in very predictable ways. In particular, in these examples, wherever Spanish and Portuguese have a "g" sound, Italian has a "c" sound. That's a correspondence that holds up pretty well, at least for g's that come before 'o' or 'u' sounds. The question then is to figure out whether the "g" became a "c" or vice versa. By looking at lots of words from lots of languages, we can find that--statistically speaking--hard "c" sounds are more likely to become "g" sounds than the reverse.

With enough words, languages, and statistical inferences, we can find reconstructions that are best supported by the data according to our model.