your PC has this IP address, it is a magical thing given to your precious PC or other device connected to internet at your home, or pretty much any other place, when you connect to a PROXY server, the mr. proxy this magical dwarf hides your IP from the big bad man at ISP or your own router, yes he might live in your router, want to block your Pr0n or some other very important content on the internet that you want to watch.
So your PC gets internet from your internet provider, but instead of going straight to *insert sample pr0n website* it goes to proxy and asks to fetch the *insert thesame sample pr0n website* so the internet provider wouldn't know you were going to *insert sample pr0n website* or even the *insert sample pr0n website* wouldn't know you were there, as it would thing its the proxy doing the bad thing.
If that doesn't help, im out. can't think of any way to dumb it down anymore without firing up paint. But no i don't think you could use a Proxy to access the JPN PSN. Why can't you do it normally anyway? Japan hasn't IP banned you has it them whatever?
Your explanation is slightly off, and can be made even simpler.
Correction: Your local network router/modem assigns IP addresses (like postcodes) to each device connected to it. Each computer, Xbox, PS3, Wii, etc, will all get their own IP for use in that local network. That IP is NOT for the internet, it's your your localnet. For the internet, only your router/modem connects to the internet, so only your router/modem gets an internet IP address (assigned by the ISP). You get one public IP address, and any number of private/local IP addresses.
Proxies for noobs: Proxies are redirect sites (or IP servers) that take any URL request for you and sends you the website through them.
Simplified version: You (person A) wants to ask person B a question, but don't want anybody to know it's you asking. You ask person C to ask for you. Person C asks person B, then comes back to you with the answer. As far as person B or any onlookers know, only person C was involved because you never directly contacted person B.
EDIT:
Person C is your proxy in asking the question, and that's exactly what a proxy site/server is for. I could've twisted the example in talking about 12yo kids trying to buy an R18 porn game by getting their older friends to buy it for them, but that's illegal and proxy sites are not illegal.