You cannot select part of an image like you can in Photoshop. Illustrator treats the image as an object. Think of the image acting like a layer in Photoshop and Illustarator’s ability to edit the image is limited to about what you can do in the Layers panel. Since you can’s select part of the image you have to find another way to interact with it.
As others have pointed out, a clipping mask is he best option. Essentially this is a path that acts as a window, showing only the parts where the path overlaps the image. That path can be any shape you draw, including circles, rectangles, stars, or text.
As all you want appears to be a rectangular mask, just draw a rectangle, select both the image and the path, then go to Object > Clipping Path > Make.