Why do ribbons curl when we stroke them with scissors? I have recently learnt how to make quilling swirls [also called paper filigree] ,one of the methods to curl the paper strips is to quickly run your fingernail on the underside of the strip you want to curl.
This also works for curling decorative ribbons on gift hampers with scissors.[see Method 1 in the wikiHow page].
So, I have searched online for the physics behind it curling and found none which are actually related to curling with scissors. Most posts dealing with printed paper or wet paper curling due to moisture and subsequently swelling by absorbing moisture.

*

*Why do ribbons curl?

*And why  they curl only downwards towards the side where we use the scissors?

 A: This is out of my reasoning so I can't be sure.
When you slide the blade over the surface of the ribbon, the friction forces whatever is quasi monodimentional in the latter to be stretched in the sliding direction.
This can applies to long single molecules as well to fibrils and fibers they may form. The stretching involves only one side of the ribbon, which therefore bends
Well known analogue is what happens in bimetallic strips https://en.m.wikipedia.org/wiki/Bimetallic_strip
What I am sure of is that a polymer having low dimensionality deposited on a substrate can be effectively oriented by simply pressing and sliding a smooth body over it. So the mechanism could well be that I propose above.
edit after watching the clip linked by another user in the comments to your question. Indeed  the curling is due to the different stretching of the two sides (or better layers beneath the two sides). However, the set up does correctly reproduce what happens in curling with a scissors. The side which is stretched is the one on the outside so the ribbon correctly bend towards the blade, i.e. the reason for the bending is geometrical rather than the direct blade - ribbon contact.
The latter works if you let a flat ribbon on a flat surface and slide the blade over it and all is kept straight.
I just tried it and indeed the curling is now in the opposite direction (tough the radius of curvature is bigger in the usual way. But this is just matter of the set up, I didn't want to ruin my scissor by putting much pressure on it).
