Their impact is because they are the lowest cost producer in the world with the largest proven reserves, and they can decrease and increase output whenever they want. Their oil fields are paid for, they are not dependent on high oil prices to run their country, the govt owns their oil fields and 100% of the production apparatus, and their location allows for easy exportation of the oil.
The Saudis are currently benefiting from low cost oil by bleeding Russia and Iran, and discouraging further investments in shale oil and deep sea drilling. The US and Russia both produce almost as much as or slightly more (depending on source) than Saudi Arabia, but their production costs are much higher. The high investment costs of shale oil, and Russia's extreme dependence on petrodollars prevent ramping of production to control prices.
Their attempts have backfired a little because it does appear as if they lost control a bit of oil prices...I doubt they wanted them to slide so far down, but they can live with prices of $40 according to their statements. The activities of commodity traders also contribute to the wild swings in oil prices both at the low and high end. It also looks like most of the other OPEC members don't want to go along with a production cut because they need the $$$, and the Saudis don't want to cut alone so for now we have cheap oil!