There are two ways that oil and gas activity can cause earthquakes. The first is the use of hydraulic fracturing (“fracking”), the high pressure underground explosive activity that can itself provoke lands to shake. The second is the use of waste injection wells, which can trigger geologic faults to shift. Both activities can produce earthquakes which then can cause damage to property as described in this interesting article:
http://www.wsj.com/articles/frackings-new-legal-threat-earthquake-suits-1427736148

When operators use fracking techniques they “shoot” chemicals into the ground at extremely high-pressure. The high pressure can stress cracks that lead to minor earthquakes. A much worse problem related to fracking is the use of injection wells, which can cause earthquakes. Oil and gas production activities generate large amounts of contaminated wastewater. There is no way to deal with this waste without some sort of risk to public health and the environment. Importantly, the recent increase in use of horizontal drilling and fracking has significant implications for injection well use. Fracking extraction and production (E&P) techniques generate exponentially more wastewater than traditional types of E&P activities. Typically in Louisiana the waste is discarded by injecting it at high pressure back into the ground into large underground formations that were previously subject to oil and gas extraction. Oilfield operators have been doing this for decades. But it is also widely known that the uncontrolled use of injection wells has unintended consequences.

The high-pressure injection of wastewater into a waste injection well has the effect of lubricating underground faults that are already under significant fault stress. Often, this lubricating effect causes formations to shift, causing earthquakes. Awareness of earthquake problems associated with injection wells dates back decades. In the 1960s in Colorado, the U.S. Army stopped using injection wells because they believed it was causing earthquakes. More recently, studies have found that where injection well use ramps up, earthquakes have become a major problem. For example, after Ohio became a destination for fracking wastewater injection, the state froze operations at several injection wells after heightened seismic activity. Similarly, in Oklahoma, the state had averaged about 50 earthquakes per year until 2009. Then in 2009, as fracking ramped up, the number of earthquakes jumped to over 1000. The Ohio and Oklahoma examples are but two of many.

The problem is not easy to solve. Most engineers feel that injection of oil and gas waste is the most environmentally sensitive way to dispose of that waste. But there is no perfect solution to the waste problem, which is made much more daunting in the context of fracking. The Safe Drinking Water Act exempts fracking as a result of the 2005 Energy Policy Act (the exemption is commonly known as the Halliburton Loophole). But the EPA still retains power over threats to groundwater, even if that power is largely unused.

The implications of the earthquake problem are many. While it may seem easy to dismiss small earthquakes as unimportant, oil and gas wells are so brittle that even small jolts in the faults can cause cracks in well-casings that lead to heavy contamination of groundwater and soil. The same can be true of water wells. For this reason, earthquakes may often be small enough such that they do not topple homes and businesses, but that does not mean they cannot be potent to human health and the environment. In any event, the lubrication of faults can hasten larger earthquakes.