You can see through the door because the holes in the metal mesh are small enough for the light“waves” to pass through, but the microwaves are too big to fit through the holes (I know this sounds odd because we are talking about something we can’t see!).
Just imagine that you have got a wire mesh, with holes about the size of a pea in it, along comes some light (imagine it as a marble) that’s 1/10th the size of the hole–it will drop through a hole in the mesh, now along comes the microwave which is a bigger glass marble, about the size of a 2 pence coin, it’s far too big so it just bounces off the mesh, it can’t get through.
This is just what happens with light and microwaves in the microwave. To the microwaves the mesh looks just like a solid metal screen; and they bounce off just like radar waves bouncing off of a ship or aeroplane, but to the light the screen is not very substantial and the light just streams through.