You want to try a couple things.
1.) select all and then hit shift m (on a mac, I dunno on a pc but its for the "shape builder tool).
while holding the option key, hover over individual points in the shape/ design. The strokes that appear in red are whole segments you can delete. Just click while holding option to delete.
If that doesn't work
2.) Any path can in Illustrator can be converted to a shape and built on to or knocked out of other shapes. Now with the shape selected navigate to object > path > outline stroke to make the stroke an object.
Now with both objects selected, choose the 'minus front' option from the pathfinder panel to knock out the front shape. This can be particularly useful when designing logos where you want to show through to the background that the logo is overlaid onto.