When something moves very quickly, you can't see it clearly - it's a blur. ...and a dog has fur. :)

Thank you both very much for your explicit explanation

Natalia is right. In English, if something is 'blurred', it is not a clear image. Your sight can be 'blurred'. A photograph can be 'blurred'. So, if something happens 'in a blur', it happens so fast that you can't see it clearly. The writer uses the phrase 'a blur of fur' to suggest that the dog was moving so fast that someone watching would see a 'blurred' furry image but would not see the dog clearly.

It means "really fast", I think.