Yeah, the file was obviously originally created in a program that can't process a true gradient, so it splits the colour band in to heaps of tiny strips, each incrementally changing from one colour to the other.
The long-winded process I would follow would be:
* Lock all the items that are correct, ie, everything except those two shapes (Object - Lock Selection)
* Select the two shapes, and go Object - Clipping Mask - Release
* Still in wireframe, Ctrl+A (select all).
* Now while holding shift, try to click on the edges of just those two curved shapes. So holding shift, click on one of them, then the other. You should then have just the tiny colour strips selected, so hit Delete.
* It's likely those two shapes will be invisible in Preview mode (as they were simply a mask, not an object unto themselves), so assign them just a simple solid fill, any colour.
* Object - Unlock All. You can now select everything.
That should be all you need.