An interesting concept. The idea that Dawkins wishes to convey, I think, is that in actual fact, all the human-scale phenomena that we notice - our behaviour, our thoughts, our rationalisations and so on - are actually the outcome of the way our genes behave. And they are "selfish" in the sense that what a successful gene will do is ensure its propagation in increased number.
But we rapidly come up against problems in the way in which we use language. "Selfish" is an expressive word - but it is fundamentally misleading. We have an image of what "selfishness" means - it means grabbing resources for oneself to the exclusion of other people. Well, that's what a gene which will increase in abundance will do. But the word "selfish" is an anthropomorphism - and Dawkins is trying to tell us that such human behaviours don't really exist, but are the outcome of the behaviour of our genes. In any case, our genes aren't conscious entities - they don't choose to behave in a way that grabs resources; it is simply the case that a gene will only propagate into the next generation if it does something that encourages its survival.
There are other issues with this, of course - it is misleading to think that one gene in a complex organism has a direct influence on its abundance in the next generation.
So despite the power of the image, selfishness is one of the darwinists' famous "invisible pink unicorns" - a mythical creature that, even if it existed, would be invisible.