Summary: It's not water itself that is burning. Methane dissolves in the water, and the methane burns. However, there's no indication that this methane is caused by fracking (as opposed to other causes)
See Tap Water Torches: How Faulty Gas Drilling Can Lead To Methane Migration where Pennsylvania State University geologist Dave Yoxtheimer explains the process:
the gas can escape, and migrate up to the surface through faults and water wells
the article further explains:
Tracing the source of stray gas appearing at the surface, can be complicated and mysterious. Investigators often use isotope identification to get a “fingerprint” of the gas. But just like in law enforcement, fingerprints can be tricky. In the case of Dimock, Pennsylvania, the vertical wells drilled by Cabot Oil and Gas were fracked. But the Pennsylvania Department of Environmental Protection did not connect the flaming water taps to fracking, they blamed poor well construction and over-pressurization. When a wellbore is drilled, a steel casing is sent down the hole. Then cement is poured down, and pushed upward to seal the open space between the steel casing, and the rock. That cement seal is supposed to prevent any gas, or fluid, from migrating into or out of the wellbore, and from using that space between the rock and the newly drilled well as a conduit. But that cement job doesn’t always work. And in the case of Dimock, it failed miserably.
Fred Baldassare worked as an inspector for DEP [Pennsylvania Department of Environmental Protection], where he spent six years investigating methane migration and helped out on the Dimock case. He was the guy who looked at the geochemistry of those flaming taps. In other words, he looked at the gas fingerprint. Baldassare says the evidence linking the flaming tap water to gas drilling by Cabot is overwhelming. But where that gas actually came from, whether it was from deep in the Marcellus formation, or whether it was from a more shallow formation, is unclear.
“The gas was nearly an exact match to the gas coming from the Marcellus wells,” said Baldassare. But we couldn’t say it came from there because there are gas deposits above the Marcellus that have the same fingerprint.”
Baldassare says drilling, along with a bad cement job, can cause any gas pocket that has been stable for thousands of years, to start moving. That’s because methane, under high pressure, wants to go to an area of lower pressure. And drilling, whether it’s a vertical well, horizontal well, deep well, or shallow well, can provide that opportunity. But other things can too, such as coal mining. It can also happen naturally. And that’s where that fingerprint comes in. Baldassare says he spent a lot of time tracking methane migration before the first Marcellus Shale gas well was even drilled.
“There are examples of [methane migration] throughout the northeast that have nothing to do with gas activity,” says Baldassare. “And there are others that do happen because of drilling or mining activity.”
So in conclusion:
Methane coming out of solution from tap water burns (tap water meaning well water)
In some (but not all) cases this is due to gas drilling, but not fracking
For peer reviewed articles see Natural gas: Should fracking stop? Nature 477, 271–275:
Another peer-reviewed study looked at private water wells near fracking sites [reference 4]. It found that about 75% of wells sampled within 1 kilometre of gas drilling in the Marcellus shale in Pennsylvania were contaminated with methane from the deep shale formations. Isotopic fingerprinting of the methane indicated that deep shale was the source of contamination, rather than biologically derived methane, which was present at much lower concentrations in water wells at greater distances from gas wells. The study found no fracking fluids in any of the drinking-water wells examined. This is good news, because these fluids contain hazardous materials, and methane itself is not toxic. However, methane poses a high risk of explosion at the levels found, and it suggests a potential for other gaseous substances in the shale to migrate with the methane and contaminate water wells over time.
Where reference 4 is Methane contamination of drinking water accompanying gas-well drilling and hydraulic fracturing Proc. Natl Acad. Sci. USA 108, 8172–8176 (2011)
average and maximum methane concentrations in drinking-water wells increased with proximity to the nearest gas well and were 19.2 and 64 mg CH4 L-1 (n = 26), a potential explosion hazard