It was a part of a standard Singer accessories set that came with every machine, until some time in 1930s. Then it was no longer required as the fashions have changed.
This is what it is for: to make braided designs. The underbraider makes it as easy as stitching along a dotted line (not necessarily all that easy if the dotted line meanders in tight hairpin turns!).
And this is what it looks like and how you fit it. Use Singer’s quilting foot without the guide, or a similar appliqué foot – it needs to clear the braid channel without pressing on it, or the braid will be stuck.
Use the soutache braid, a.k.a. Russia braid, the softer the better. The one I had here was pretty stiff, I’m sure the results would be better with a silkier braid.
Then you trace your design on the wrong side of the fabric and put it under the machine the wrong side up. Follow your design and don’t worry about the braid – it gets sewn on the underside as if by magic! Use short stitches (1mm) for best results.
It comes out a bit crampled but gets ironed out very nicely.
But here is a pitfall: the braid will drag out all the dirt from the feed dogs! It’s a proper feed dog floss, it is. Not great when you’re working with whites! So, clean the feed dogs really well before beginning, and do a bit of practicing first, just to be sure. The silver lining is of course that you get a really clean machine after that! 😀