They stole my music but they gave me my name.
– Muddy Waters
What Waters says here is what I think the Rolling Stones brought to the Blues. They are, as I said before, huge fans of Blues music, and so they decided to cover those songs they love so much to share this love.And in doing so, they became famous by playing these particular songs, they became famous, but at that time, in England when you said the name Muddy Waters, close to nobody knew who you were talking about.And because they gave credits to these singers they covered they finally could obtain the recognition they diserved for the work they had done. Muddy Waters was recognize as a really great Bluesman partly because the Rolling Stones covered him and loved his music and we can’t take that away.
“What’s the point in listening to us doing `I’m a King Bee’ when you can hear Slim Harpo doing it?”
– Mick Jagger
What Mick says here is that they did not want recognition from these songs but if so people should give credits to the original singer. And that they too wanted to spread to the world these culture of the Blues music as much as they could because they loved it and they did not want it to disappear like this.