It's a huge pile of code that lets you checkpoint your virtual environments — in the same way you do "suspend to disk" or "hibernation" on your laptop. This is cool, but what's the use of it for VE? Here it is: you can actually restore ("wake up") a VE on a different machine! The process is called live migration; newer vzctl will have a tool called vzmigrate for that.
Imagine you want to add more RAM into your (physical) server, or upgrade the kernel. Such operations require a reboot of the physical server — means you have to shutdown all your VEs. But now you do not need to do that — instead you live migrate all VEs to another OpenVZ box, when do that maintenance and live migrate VEs back. From user's point of view, this looks like a delay in processing, not as any downtime. All the network connections and all the other stuff is preserved — that is why it is called "live" migration.
Another scenario is when your VE has grown considerably and it needs more CPUs than the server has. Again, you put another server with OpenVZ installed, and live migrate your VE to it. No hassles, no reconfiguration, no downtime.
Isn't it amazing? Yes it is. The most amazing thing though is the technology itself is free software. Free as in freedom, not as in beer, please do not mix those.