MSTs appear in network design, clustering, and approximation algorithms. If you are building a physical network (roads, cables, pipelines), MST gives you the cheapest way to connect everything. In clustering, MST edges with the largest weights are often cut to separate clusters.
This is the single-linkage clustering approach. MST algorithms are also used as subroutines in approximation algorithms for NP-hard problems like the Traveling Salesman Problem.