Why does the Brexit cause a fall in crude oil prices? Oil prices have also fallen sharply in the wake of the referendum
  outcome, with Brent crude down 5.2%. The price of Brent crude fell by
  \$2.68 to \$48.24 a barrel, its biggest fall since February. At the same
  time, US crude was down 5.4%, or \$2.69, to \$47.52 a barrel.


From http://www.bbc.com/news/business-36611512

But why does this happen?
I'd like to extend Lasse's excellent answer.

Fear and uncertainty are driving markets - but they can drive prices in either direction of course.

Specifically what's happening here is that the oil markets are pricing in at least two effects.

Firstly, the UK is an oil producer, and Sterling's slide means that its oil just got cheaper for other countries. This is a small effect, but relevant, as UK short-run marginal extraction costs are close to current prices, so some UK fields are marginal producers and thus price-setters.

Much more importantly, the markets are now pricing in the possibility of a further slump in oil demand, as a result of poor general economic performance. That would be driven not only by a soon-to-be-declining UK economy, but also driven by a (much larger) destabilised European economy; and also driven by the possibility that this vote will mark the turning point, where globally we move away from a consensus on the value of tariff-free international trade, and back towards protectionism and isolationism, which would leave the global economy worse off, as well as leaving most participants individually worse off too.