How do we know that these radio bursts are from billions of light-years away? NASA just announced that they detected the first radio bursts from outside of our galaxy.

Astronomers, including a team member from NASA's Jet Propulsion Laboratory in Pasadena, Calif., have detected the first population of radio bursts known to originate from galaxies beyond our own Milky Way. The sources of the light bursts are unknown, but cataclysmic events, such as merging or exploding stars, are likely the triggers.
The new radio-burst detections -- four in total -- are from billions of light-years away, erasing any doubt that the phenomenon is real.

If we don't know what the source is, then how do we know how far away the source is? How can we tell how far the light waves have traveled?
 A: 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).
A: A fundamental distance in the universe is the cosmic proper distance. Unlike luminosity and angular diameter distances which corresponds to the angular size, the cosmic proper distance is the length of light path from the source to observer. This proper distance can be calculated from the redshift measurements of the FRBs. Also the distance-redshift relation derived from the dispersion measures (DMs) of the FRBs with measured redshifts.
Conditions required for a probe to measure the cosmic proper distance are as follows:

*

*It should change with red-shift in a well-understood way and be independent of cosmic curvature.

*It should record the information on the expansion of the Universe.

Note that, standard candles and standard rulers cannot be used since they depend on the cosmic curvature.
A radio signal traveling through IGM exhibits a quadratic shift in its arrival time as a function of frequency, which is known as the dispersion measure (DM). This DM can be measured with high accuracy for an FRB. However, this DM includes several components that are caused by the plasma in the IGM, the Milky Way galaxy, the host galaxy of the FRB, and even the source itself. $$DM_{obs}=DM_{IGM}+DM_{MilkyWay}+\frac{DM_{host}+DM_{source}}{1+z}$$
Only the $DM_{IGM}$ contains the information of the proper distance. Other components therefore need to be eliminated from the $DM_{obs}$. The $DM_{MilkyWay}$ can be easily calculated from pulsar data and can be eliminated.
The $(DM_{host}+DM_{source})$ can be determined based on the assumption that it doesn't evolve with redshift and fortunately the last term in the above expression further decreases due to the $(1+z)$ term in the denominator. While $DM_{IGM}$ increases with redshift, $\frac{DM_{host}+DM_{source}}{1+z}$ is not important at high redshift and can be treated as an uncertainty.
The DM of radio signal is proportional to the integrated column density of free electrons along the line of sight. In addition, the redshift measurement of the source gives information on the expansion of the Universe.  For an FRB, the DM can be measured directly and its redshift can be estimated by observing its host galaxy or afterglow. Therefore, the distance-redshift relation can be derived with the DM and redshift measurements of a large sample of FRB data and ultimately the proper distance of the FRB can be calculated.
