Hardness is really the simplest term involved in this thread.Forgive me if someone's already answered this, but "hardness" is actually a pretty complicated and fuzzy term. This Knife Steel Nerds article gets into some of the details. Extremely roughly speaking, you could say hardness is the ability to resist plastic deformation. Obsidian, for example, is extremely hard. No matter how much you hammer on a chunk of obsidian you're not going to squish it into a different shape. However, it is also brittle, meaning it's fairly easy to chip/shatter it, which is different from deforming it.
So you want a hard edge so it doesn't "squish," but you also want a tough edge so it doesn't shatter/chip like obsidian.
I've made some knife scales but I've never made a blade so I won't attempt to say I'm a knifemaker. But what I am is a structural engineer and I've done some study and research into fatigue and fracture.
So back to hardness- the primary property of steel is its yield strength. All of us have taken a piece of wire and bent it. You bend it a little bit and let go, it straightens back to its original position. You bend it more and it keeps a permanent deformation. The point at which it starts to permanently deform is the yield stress limit. You bend it a lot more and it deforms further, or if it's a high strength wire maybe it eventually breaks. How far it deforms before it breaks is its ductility. The point at which it breaks is its ultimate strength, or fracture strength. Very high strength steels can never have an actual yield point, they just get to a point where they can fracture. If instead of a wire we had a bar and we pushed on it with a pointed object- at the point where the pointed object leaves a mark is related to the yield strength. Take a steel with higher yield strength and it will be harder to leave a mark on it so we think it is harder steel. So hardness is nothing more than a measure of the yield strength. I should say if you throw a lot of carbides in there it can make the steel seem harder without increasing its yield strength.
Now remember what we said about ductility- toughness is mostly a measure of ductility. Also, despite the old fashioned test used to measure toughness, toughness has nothing to do with impact. Somebody 100 years ago could have just as well come up with a different way of measuring toughness without using impact.
How to relate this to edge stability- whatever you are putting the edge against, at whatever speed, puts various types of stresses in the steel. There will likely be tension stress- if that point of the steel reaches the yield stress limit then the edge rolls. (You roll it far enough it can eventually fracture.) If it's a steel that is higher strength (harder) but doesn't have a yield point, you have to put more force on the edge to get the stress higher but eventually you reach a point where the tension stress reaches the fracture point and the edge chips. So the rolling point for a less hard steel occurs before the fracture point for a harder steel. This is why hardness gives higher edge stability.
If you want to discuss the entire knife and not just the edge then toughness can come into play in other ways. Blades can have saw teeth, choils, holes, corners where the tang necks down, notches for locking mechanisms, lots of places that cause stress concentrations. If a blade has low ductility (toughness) then when the stress at one of the stress concentrations reaches the fracture stress then the blade fractures. If a blade has lower strength but more toughness, when this point in the blade reaches its yield limit it just yields a little bit and doesn't initiate a fracture. This is the advantage of toughness in a blade as a whole.