In this example of edge detection:

http://www.mathworks.com/help/images/edge-detection.html#buh9ylp-13

How would I remove the gray background in the grayscale image at the beginning?

 Answer: 

Removing the background befor you apply an edge detector makes no sense as the more or less homogeneous low frequency background will result in a zero gradient anyway. It won't improve the results of your edgedetection.

If you only want to cut out the coins to put them on a different background simply apply a global threshold

Remove thos tiny speckles with some morphological operators and blur the mask so you won't have fringed coin edges

Apply some logical operations to mask the original image