Those are the line numbers and the operation to change one to the other. "5c5" means "line five changed (replaced) to line five", "7d6" means "delete line seven", "21a22" means "add (append) line 22"
You can change the output format using options such as --context=NUM or --unified.
Here is some more information on output formats from the info file for diff.
This command would only show the lines that are different without showing the line number and modification type (it will also eliminate the "----" dividers):
diff cookies.old cookies.new | grep '^[<>]'