This looks like a bitmapped image. Am I right? If so, you'd need Photoshop to edit it. Illustrator is ideal for creating/editing vectors, but raster stuff... not so much.
There are two possibilities.
If this is a bitmap, you can select it and under the Object menu choose Crop Image. This command will delete anything outside the cropping area.
If this is vector, or a combination of bitmap and vector, you can select all the objects. Object > Group them for convenience. Then draw a shape that is the size you want for the crop. The shape must be on top of the other objects. Then select everything and choose Object > Clipping Mask > Make. This command will hide the area outside the clipping mask, but will not delete it.
Assuming the "unwanted parts" is anything outside the logo?
If so, make a copy of the logo vector, paste in place, select it and the backgound graphic, right click ---> create layer mask