Imagine a virus spreads between individuals when they are in the same room. Then, consider the hypergraph where each hyperedge is the set of individuals who happened to be together in a same room. It may represent the network on which the virus spreads (although adding time information seems needed).
More generally, since hypergraphs are equivalent to bipartite graphs, they model many real-world situations.
Another example is the co-authoring hypergraph, in which each paper leads to an hyperedge between its authors. It is used to study the structure of scientific fields.
In recommender systems, one also considers hypergraphs over products, where hyperedges are sets of products bought by a same client.
The wikipedia hypergraph page has some pointers to various applications.
I hope this helps, good luck with your course!