What is happening is when you trace your image, Illustrator is making your "transparent background" into "white space". What you need to do is remove this from your trace.
1.Menu > Window > Image Trace
2.A new window will open. This will give you a lot of trace options
you can play with and the one in particular you are looking for is
under the advanced drop down.
3.Check Ignore White Space
You will now have your cutout object. If you are on a Mac use Cmd+Shift+G etc.
Another way to achieve this would be to expand your object and delete it manually. To do this you need to do the following.
1.Menu > Object > Expand - Click ok.
2.Ctl+Shift+G (ungroup) You may need to hit it a few times.
3.You may need to right click on the object and select release clipping mask.
4.Click the white area and delete.
5.Regroup object by selected the whole thing and pressing Ctl+G (Group)