Scaling on the backend
In a very simple setup, one DNS entry goes to one IP which belongs to one server. Everybody the world over goes to that single machine. With enough traffic, that's just too much to handle long before you get to be YouTube's size. In a simple scenario, we add a load balancer. The job of the load balancer is to redirect traffic to various back-end servers while appearing as one server.
With as much data as YouTube has, it would be too much to expect all servers to be able to serve all videos, so we have another layer of indirection to add: sharding. In a contrived example, one server is responsible for everything that starts with "A", another owns "B", and so on.
Moving the edge closer
Eventually, though, the bandwidth just becomes intense and you're moving a LOT of data into one room. So, now that we're super popular, we move it out of that room. The two technologies that matter here are Content Distribution Networks and Anycasting.
Where I've got this big static files being requested all over the world, I stop pointing direct links to my hosting servers. What I do instead is put up a link to my CDN server. When somebody asks to view a video, they ask my CDN server for it. The CDN is responsible for already having the video, asking for a copy from the hosting server, or redirecting me. That will vary based on the architecture of the network.
How is that CDN helpful? Well, one IP may actually belong to many servers that are in many places all over the world. When your request leaves your computer and goes to your ISP, their router maps the best path (shortest, quickest, least cost... whatever metric) to that IP. Often for a CDN, that will be on or next to your closest Tier 1 network.
So, I requested a video from YouTube. The actual machine it was stored on is at least
tc.v19.cache5.c.youtube.com. Those show up in the source of my webpage I'm looking at and were provided by some form of indexing server. Now, from Maine I found that tc19 server to be in Miama, Florida. From Washington, I found the tc19 server to be in San Jose, California.