You can do it this way:
Open the drawing, and choose File - Find References. Copy Files, and choose the option to Flatten to a single folder. Make sure to also check the box to include Toolbox components if you used any.
Then you just set a path to a new folder, and all of the parts are magically copied for you!
Thanks Fred, it does copy all info. But it doesn't combine the drawings into one package. (like the picture I attached)
Actually, it looks like the File Find References tool just leads you to the Pack and Go dialog.
There is another way to accomplish this task with the SolidWorks Explorer program, but I never install it, so I can't see the steps to follow right now.