How to remove/delete black color around image (or make transparent)

If there is a clipping mask or compound path preventing you from selecting the black backround, you can double click on the Illustration to enter isolation mode and delete it.

Open the layers panel. Most probably the black in just a black rectangle in the background of the image (which is at the bottom of the layers panel). It might be locked. Unlock and delete.

If that does not solve it, please show a screenshot of the layers panel.

Hit A to switch to the direct select (AKA: hollow or white)  arrow tool and do a Command Y to see in key lien view who this is constructed. Command Y again to go back to preview. Select what I suspect you to have an outer rectangle and delete.

To confirm transparnecy do a shift command D to show transparency. Hit that key compound again to go back to normal view. This will help you with the white outline around the elephant.