Please try to show us your Layers Panel with (visible and selected) checkered background layer (if exists)
I got rid of the dark backgoround only for it to turn white.
Got rid of the background all together but the ornaments are no longer transparent.
View> Hide Transparency Grid, will turn off the checkerboard. The transparent areas will become apparent when you place the file on top of other elements (in InDesign), or you place objects below it in Illustrator. It appears the ornament includes transparent areas so you can simulate a reflected object.
That one won't go away when you hide the transparency grid.
Those are rectangles filled with color on the bottom layer of the document.
Please see this video: https://youtu.be/Tiqm5hdQA5A