I gather that the equilibrium constant K = 1 implies that, at equilibrium, neither the forward nor the backward reactions are thermodynamically favoured.
There is nothing special about K = 1. At equilibrium, no matter the value of K, there is no net reaction. Forward and backward reactions proceed at the same speed.
But if pressure of the system increases and this shifts the position of equilibrium, the value of K remains to be 1.
Yes, even a reaction with K = 1 can react. Just imagine you start with some reactant and no product. This situation is not at equilibrium. The reaction will proceed in the forward direction until Q = K. Then, you can disturb the system, changing Q, and there will be a net reaction again, changing concentrations until Q reaches K again.
Please can I have a simple and comprehensive explanation of what the value of K actually tells us?
You said "please", so I will try for a short, simple and comprehensive answer. The value of K allows you to calculate equilibrium concentrations, no matter which concentrations you started with. For a lower K, the product equilibrium concentrations will be lower than for a higher K (same initial concentrations). The opposite is true for reactant equilibrium concentrations.
It should be clear that the equilibrium concentrations for a given K are not always the same. The simplest example is doubling initial concentrations (while keeping the volume constant). If the equilibrium concentrations were equal for both cases (initial concentrations doubled or not), it would go against the conservation of matter. (We doubled the amount of species present initially, so we can't have the "old" amount of equilibrium species).