For example, by removing the longest edge in the MST, one obtains two groups, also called clusters. This may be useful for clustering objects. Biology and science in general is preoccupied with grouping objects, plants or animals. We refer to this as classification or taxonomy. This yields the grouping shown in the mammal's milk example.
Proof is by contradiction.
You have a partial solution S, which consists of several connected components. Add the shortest bridge so now you have exactly one cycle with at least 2 bridges. So, again, you have a tree but now of shorter length. In class we were asked to to show that each node is connected to its nearest neighbour, this is what we have proved in the graph found above. This example may be generalized to every node.