Hypergraphs can arise as Bruhat-Tits buildings of groups, see e.g. [here][1].


Some real world applications:
[In this article][2] the authors list some applications to biology. Their nice starting example is that if one wants to model a chemical reaction one can write A-->B for a process which transforms A into B and see this as the edge of a graph. Sometimes such a process only works in the presence of some catalyzer (A+C-->B+C), making it a relation between three instead of two ingredients and giving a 2-edge of a hypergraph.


  [1]: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.dmj/1184341240
  [2]: http://www.ploscompbiol.org/article/info:doi%2F10.1371%2Fjournal.pcbi.1000385