OK, it is painfully (heh) obvious why cats have claws: to climb trees, for helping catch prey and for defense. What I can’t figure out what purpose dogs’ claws serve. All they seem to do is make a racket on the hardwood floors, and need constant trimming.
Among other things, it gives them traction when running. Think cleats.
You could ask why we have fingernails; because an ancestor had them, and there’s no evolutionary reason not to have them. I would imagine that they can be useful, even in modern dogs, for catching prey and defence (I’ve known some dogs to use their paws a lot in play).
If you’ve ever gotten scratched by a dog, you’ve surely learned that they’re a bit better defensively than our finger nails. We leave shallow scratches, their claws leave welts and/or draw a little blood.
digging.
scratching themselves for relief?
In the wild, canids’ claws would be worn down from digging, fighting, and general running. Some, perhaps many, domesticated dogs enjoy digging, apparently as a hobby.
It helps them fight. Ever wrestled with a upside down dog? They rake you with their claws and snap their jaws from side to side.
And some, Jebus bless them, see it as a raison d’être. 
OK, that’s good, plus digging too. Although you can get scratached playing or fighting with a dog, I still think their teeth are the primary defense, and anything from the claws is just accidental.
Years ago I had a Borzoi who ran with me every day on hardtop roads, and I never had to clip her claws, just nicely worn down. 
I guess these are all good answers, so will stop trying to produce a genetically clawless dog.
I always thought it was traction for running. If you look at the paws of Cheetahs, they have claws that more closely resemble canines than cats, which figures since running around plays a lot more into their own lives.