I do understand how DNS works, and I was also confused about why changing your DNS server might affect download speeds. On the surface of it, it shouldn't -- it doesn't matter how quickly I look up your address, if the road to get to you is slow, I should have slow performance.
Then I remembered something I read about Google and YouTube bandwidth -- and it boils down to this:
* Google (YouTube), Apple, and other providers of big content (or CDNs) may have reached an arrangement with your ISP to offer a more direct connection to their content, perhaps over dedicated links that reach more directly to the content servers.
* The easiest way to inform your computer about this more direct link is to send you an address specially for the purpose -- if you are on Verizon FiOS, then the Verizon DNS servers will send you the correct address for Verizon's fast connection to Apple.
* If you don't use Verizon's DNS, but, for example, use 8.8.8.8 (Google), there is no way that Google would know or should ever send you the specific Verizion-Apple IP address.
I had a similar configuration myself and after removing the 8.8.8.8 Google DNS and using the default Verizon DNS, I went from 300kbps to 6mbs.
So you can stop yelling at each other, you're both right: DNS can't actually make a difference in download speed, but it could send you to a private wormhole-type IP address that actually gives you much faster download speeds because of the physical topology of the network and agreements between carriers and Apple to facilitate faster speeds. Which, at least in my case, it seems to do.