Q: Where does Apple Mail store attachments?

Answer

The Attachments folder is located in your user home directory in one of the following locations (depending on what version you're running):

Users/Username/Library/Mail Downloads

Users/Username/Library/Mail/Mail Downloads

~/Library/Containers/com.apple.mail/Data/Library/Mail Downloads

Users/Username/Library/Containers/com.apple.mail/Data/Library/Mail Downloads

If you can't see your Library folder, see:
How to make your Library folder visible in the Finder in OS X 10.9 (Mavericks) or later