Typical rocket launches cost anywhere from $20m ---> $200m to actually launch. The cost to the customer, whoever that is, will likely be more if it's being launched by a private company.
However that's because, until SpaceX came along, they threw the whole rocket away with every launch. So that $20-$200m is lost in the ether. Imagine you threw away a Boeing 747 every time you flew it - flights would be extraordinarily expensive (and, in fact, we'd likely have never invented a 747 as a result). What SpaceX have done with the Falcon 9 is make the First Stage of the rocket land and then re-use it. This is the most expensive part of the rocket, but there's still a lot of it that gets thrown away.
The plan with this new one is that literally all of it is reusable. It all comes back down to earth in a way that means it can be used again - in theory very quickly (unlike the Space Shuttle, which took months and months to refurbish, effectively requiring it to be taken apart and re-assembled every time it came back). So the model then becomes a lot more like an aeroplane than a typical rocket. The fact, then, that the rocket's very large doesn't increase its costs by as much as you'd think, because again, the only thing you're not getting back is the fuel (though even that's not totally true - this rocket uses a Methane and Oxygen mix, not Kerolox or traditional rocket fuel. As such, with enough energy - say, from solar panels that Musk also makes a lot of - you can actually capture the Hydrogen, Carbon and Oxygen out of the atmosphere before spraying it back out the back of the rocket. In this sense, it's carbon neutral and potentially created using 100% renewable energy).
So I dunno if Musk's anticipated cost is accurate, but you have to think about it more like a plane than like other rockets. Also bear in mind that most of the big rocket manufacturers are, finally, catching up to the idea of reusing elements. This means that, in the "rockets-as-planes" world, we're barely at the Spirit of St Louis level, let along the DeHaviland Comet or Boeing 787 stage. SpaceX are absolutely blazing this trail, but it won't be too long before we have Blue Origin, Orbital ATK, ULA, Arianne etc all with their own "rockets-as-planes". It's an exciting time to be a massive rocket geek!