The idea is that you can use an edge that is NOT as thin at the cutting edge but that slices just as well, if not better...
That way the carbides at the very edge are more supported and never break off, dulling the edge...
A true convex almost never has to actually be resharpened, just a quick pass or two on some cardboard or even your paints leg and you can butcher four or five deer... Afterwords you may need to use some leather with polishing compound on it, but not actually sharpen it...
I had a friend with a SAK that I would sharpen very sharp and he would bring it in a few days later, so dull you could not cut yourself...
I finally asked him, if I could do an experiment and convex it...
He said yes, and an hour or so later gave it back to him sharp enought to shave and slick free hanging paper...
It was over 2 months before it needed touching up, and that was done with some leather with green polishing compound on it!
It seems impossible, but it really works...
BTW... You know the old sharpening stones in the hardware store, that were so slope backed that you never wanted to use them??? Well, if you use one of them, and folow the curve, you will get a convex edge too.
Flat edges are(mostly) a modern invention...The Boy Scout manual promoted it, and it became the standard, but it was NOT the way that knives were sharpened before that.
That, along with convex being hard to do by machine, let to people forgetting about the advantages of a convex edge...

Another advantage of the convex is in deep slicing... Look at any sub-sonic airplane... you will not see sharp transitions...
Cutting thick material is simular to fluid dynamics in some ways...