I'm going to assume here that the first photo in your original post is the miter box you found. If that's not the case, a photo of it as it sits will help.
These miter boxes take a backsaw, like the ones at this link but longer and with a wider saw plate (the thin sheet metal part with the teeth):
http://www.leevalley.com/US/Wood/page.aspx?p=57152&cat=1,42884. The saw plate goes in the narrow slot in the saw guides, with the back going in the wider slot at the top of the guides. The guides then slide up and down in the box. Once you lube the saw guides lightly, you'll find that you can set the saw on the work, start sawing, and the weight of the saw and the guides together will carry the saw down through the work.
You can see an example of the saw in its guides in the photos here:
http://www.wkfinetools.com/hUS-borTools/MillersFalls/pubs/mf-MiterBoxes/MF-MiterBox-HowTo.asp. This site also offers a download of an instruction manual for Millers-Falls boxes. The basic principle of these boxes is like that of the Goodell boxes, although the angle locking mechanisms, saw stops (to keep the saw from going down far enough to damage the teeth on the metal), and the like will differ. Still, understanding how one box works will help you understand how yours works.
I'm not aware of any new source for these saws. You'll need to haunt yard sales or eBay, or contact one of the vintage tool sellers to get one. Shipping will add up; sometimes, shipping is more than the price of the saw itself. I've had good experiences with Brass City Records and Tools and Fine Tool Journal, but there are lots of sellers. You might also wander over to WoodNet, the hand tool forum of which has kind of a saw focus; you might find a seller on their buy-and-sell forum.
You might look at the saw guides closely. On the Millers-Falls boxes, one is for the front and one for the back; and, when I got my M-F box, their positions were reversed. Not sure what difference it makes, but I did switch them around.
My Millers-Falls box is beyond fantastic; it has the same relationship to a hardware store box that a Rolls Royce has to an arthritic mule. I still remember the day I trimmed a saw kerf's width of wood off a piece of molding that I'd cut too long. You can't do
that with a hardware store box. When these were selling at retail, they went for nearly the modern retail of a pro table saw* - and were probably just as valuable to the carpenters using them.
*That's assuming I did my inflation calculation correctly.