Since there is currently no path data to actually delete, you will need to draw the area you want to remove. You can do this with the Pen Tool, Rectangle Tool or whatever else, just make sure that you have a closed shape.
Select both the grey background object and your newly drawn window object. You can do this in the layers panel if it is easier (select by clicking the circle, not just the row):
Open the Pathfinder panel (Window → Pathfinder). The second "Shape Mode" button is "Minus Front", which will cut the top object out of the bottom object. Select this.
As you can see in the layers panel, this is now a single shape, with the "window" cut out of the background (it looks the same since it was a white shape and the artboard background is white, but it is now actually a hole—i.e transparent).
You can read more about Pathfinder and other methods of combining shapes here: