How do we know that these radio bursts are from billions of light-years away?

The relevant Science summary is "Radio Bursts, Origin Unknown" by Cordes, abstract here. Briefly, it notes that a burst of radio waves will undergo dispersion in the interstellar and intergalactic medium (ISM and IGM), the amount of which is indicative of how much matter the signal has passed through.

Some further background not given in that summary: In general, the ISM/IGM consists largely of free protons and electrons. As a plasma, this medium has a frequency-dependent plasma frequency, so different frequency components will propagate at slightly different speeds (just under the speed of light). (In fact, radio waves that are too low in frequency cannot propagate at all through the "emptiness" of space.)

The more material the signal has passed through, the more spread-out the pulse will be, with lower frequencies arriving after higher ones. Assuming the original shape of the pulse is known (perhaps it is just a very sharp spike, at least compared to how broad it is by the time we detect it), then one can take the shape of the detected pulse and figure out how much dispersion has occurred. Then one asks, "How much of this is due to material in our own galaxy?" which is answerable based on maps astronomers have constructed of the interstellar (intra-galactic) medium along various sight lines. Next one guesses (in an educated fashion) how much is due to the host galaxy of the source. The rest is attributed to the IGM, which, assuming some uniform density, yields a distance. A summary of this technique can also be found on Wikipedia; see Dispersion in pulsar timing

As it turns out, this is indeed the method used in the Science paper by Thornton et al., "A Population of Fast Radio Bursts at Cosmological Distances," abstract here, to which the NASA press release was referring. These particular events were found at high galactic latitude, meaning they were seen looking "up" or "down" out of the plane of the galaxy, rather than through the bulk of the disk, so not much of the measured dispersion can be attributed to the ISM in the Milky Way.

That second Wikipedia article defines dispersion measure. The Thornton paper reports dispersion measures for these four objects are $553$, $723$, $944$, and $1104\ \mathrm{pc/cm^3}$. After subtracting the effect of our own ISM, they conclude the extragalactic dispersion measures (including any contribution from host galaxies) are $521$, $677$, $910$, and $1072\ \mathrm{pc/cm^3}$. They then assume host dispersion measures are $100\ \mathrm{pc/cm^3}$ in all cases, subtract that off, and divide by a value for the IGM number density of electrons (in $\mathrm{cm^{-3}}$, to get a distance in parsecs). Actually, this last step is a little more complicated due to the fact that the universe has expanded over the long time those radio waves have been traveling, but the authors take that into account.

In summary, radio astronomy uses the fact that the space between galaxies is not completely empty, and that radio waves, like all forms of light, slow down in various ways when traveling through matter. This is particularly useful when the signal doesn't have sharp spectral features from which to obtain redshifts (this technique being common in optical, UV, and IR astronomy).