Vacuum expectation values are important to describe the Higgs mechanism and symmetry breaking in general.
In classical field theory, we are interested in the solutions of the Yang-Mills equations and these correspond to critical points of the action.
In contrast, in quantum field theory, we use the action to compute "vacuum expectation values" of observables. A vacuum expectation value is the average measured value in the vacuum state. The thing is that many physically interesting quantities can be expressed as vacuum expectation values through various tricks.