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]: https://doi.org/10.1215/S0012-7094-07-13913-9
  [2]: https://doi.org/10.1371/journal.pcbi.1000385