It's basically the same as you and I connect to the internet except on a much larger scale and usually with built in redundancy and lots of monitoring.
Data centers have multiple high speed communications feeds coming in from all the primary carriers such as Verizon, AT&T,and all the others, that connect to multiple primary routers. From there they will go to primary firewalls, then to smaller routers, firewalls and switches.
The routers, switches and firewalls are configured so that communication automatically gets re-routed with the loss of any link.
Everything, from the links coming in, to the network devices, to the links to external sites, are monitored 24x7 by a NOC.