Okay.. thanks for adding the screenshot.
I'll explain how to "punch holes" in the raster image below, but....
The reality is if I were working on this I would not use the raster image. I would just redraw all of the rectangles merely using the raster image as a guide. It's easy enough to create rectangles in the same layout.

Having the forms as native vector objects frees you to do whatever you want to them directly.
To work with a raster image...
In order to "punch holes" in it using Illustrator, you need to use a Clipping Mask. It's not possible to merely "erase" parts of raster images in Illustrator. The best you can do is hide portions by masking them.
- Draw rectangles (or any shape) where you want "holes" and a big rectangle which is outside the dimensions of the raster image. Here, I've drawn 4 rectangles and given them a Magenta stroke so you can see them...

- Make certain the inner, small, shapes are above the large outer rectangle in the stacking order. (i.e. bring them all to front
Object > Arrange > Bring to Front
)
- Select all the shapes you've drawn. Here, I've reverse the fill/stroke so they are all filled shapes.

- From the Menu, choose
Object > Compound Path > Make

- You should see "holes" appear in the big, outer, rectangle...

- Select this big compound shape and the underlying raster image.
- From the Menu choose
Object > Clipping mask > Make...

- You should now see "holes" in the raster image...

(The yellow-green background is used to show where there is transparency.)
With clipping mask shapes, anything which is covered by the original shape will be visible when the clipping mask is created using that shape.