The whole fridge thing is just a red herring.
I find it handy that in the IPv6 world, my ISP must give me a subnet rather than just a single IP address. I can access anything at home I need to over IPv6 from anywhere. Meanwhile, NAT is resource intensive for a firewall. It's much better to avoid packet re-writing and just filter.
But much of the benefit is for larger organizations and for the future. We really are running out of v4 addresses even while some of them are being clawed back. NAT presents it's own problems, including the already mentioned resource drain on a firewall. The same hardware filtering v6 rather than NATing v4 can handle many more machines. If you as the admin of such a place (for example, a large office) get an abuse report, instead of just your external IP address and a time that may or may not be accurate, you get an IPv6 address that uniquely identifies the probably infected PC.
When I stand up a VM at work that doesn't need to be accessible by the public, I can just skip IPv4 and use it's autoconfig v6 address. No need to be concerned with depleting the much smaller pool of available public v4 addresses.