What is the intuition behind the envelope theorem?

The envelope theorem states that the derivative of the value function with respect to a parameter is equal to the partial derivative of the value function with respect to that parameter, evaluated at the optimal values of the control variables. In simpler terms, it tells us that the change in the value function due to a small change in a parameter is equal to the change in the value function that would occur if the control variables were adjusted to keep the parameter constant.