Take, for example, a room and an ice cube as an example. Let's say that the room is the isolated system. The ice will melt and the total entropy inside the room will increase.
This may seem like a special case, but it's not. All what I'm really saying is that the room as whole is not at equilibrium meaning that the system is exchanging heat, etc. inside *itself* increasing entropy. That means that the subsystems of the whole are increasing their entropy by exchanging heat with each other and since entropy is extensive the system as whole is increasing entropy. If the system is at equilibrium it has maximum entropy already.