Fundamentally this just isn't how Illustrator, or vector applications in general, operate.
Think of vector artwork as pieces of construction paper. You can't simply erase part of a piece of paper. You have to cut it then remove the cut section. This is how you would remove part of an object or group of objects in Illustrator. Features such as the Pathfinder Panel and Shape Builder Tool can make this "cutting" easier. There are a great many tutorials for these features on the web if you search.
One other possible method would be to use a Clipping Mask to hide the portion of the art you do not want visible. This is often helpful if the artwork contains many pieces or is very complex and cutting would cause either a change in the overall appearance or be to difficult to perform accurately.
- Simply draw a shape around the area you want to keep visible
- Select that shape and the artwork
Object > Clipping Mask > Make from the menu.