I love @valerio answer, but I will give it a shoot with a very intuitive explanation.
Coarse grained entropy is an observer dependent quantity which can be though as measuring the ignorance of the observer about the system (or equivalently, the amount of information that can be extracted from the system).
For instance observing a box full of bouncing ideal gas particles, I realistically ignore the exact (micro)state of the system, but I can measure that the macroscopic properties are the thermodynamic quantities (P,T,V,etc.).
Suppose now that something happen to the box.
If the process is reversible, new microstates will become possible, but an equal number will become forbidden by the constraints given by the measured thermodynamic quantities.
If the process is not reversible, more microstates will become possible with respect to forbidden ones. In effect now there are more possible microstates compatible with (P,T,V,etc.), therefore more entropy.
As a concrete example, think about a box containing white particles on one side and black particles on the other side, with a physical divisor in the middle. The act of removing the divisor imply that (besides velocity, position, etc,) now I'm also ignorant about the color of the balls in a given volume of the box.
In summary, in absence of microscopic measurements our ignorance of the system will go up or at best stay constant.