They're most likely using a content distribution network (CDN), something like Akamai. For example, the Microsoft Download website appears to be hosted on my ISP's network, whereas pinging it from one of our servers shows it elsewhere in the world.
A CDN is not simple, you would likely need to take the advice of an expert in this field to establish one. There is no 'guide'.
You can do it using either intelligent DNS where the DNS server returns the IP of your local server each time you do a query, or using anycast where the same IP is visible in multiple locations around the world. The latter being the best way (IMO) to do it.
Your best bet is to sign up with a company that provides a CDN service rather than trying to establish your own.
█ Dan Kitchen | Technical Director | Razorblue
█ ddi: (+44) (0)1748 900 680 | e: [email protected]
█ UK Intensive Managed Hosting, Clusters and Colocation.
█ HP Servers, Cisco/Juniper Powered BGP Network (AS15692).