what the hell is this site, and how does it relate to the security of my bitcoins?
I'm not sure what the purpose of that site is, unless it's a joke. It isn't a database (that would be impossible), it's generating the pages in response to queries. So possibly an attempt at a scam, as in you look for your private key, and the holder of the list uses the generated pages to narrow down the possible addresses that have actually been used (there are 2^160 possible addresses, or 1,461,501,637,330,902,918,203,684,832,716,283,019,655,932,542,976). Since people have trouble putting that in perspective, there are only 2^63 grains of sand on all the beaches of Earth.
The total number of bitcoin addresses in use is so much smaller than the amount of possible addresses, that something like this is completely and utterly useless, and it would also be trivial to expand the number of possible addresses in the future. I laughed when I saw that website, cause I just know people who don't know the math are furiously putting page numbers hoping to find one

. You'll likely witness the heat death of the universe before that happens.
Here's some good reading about address collisions/exhaustion.
https://bitcointalk.org/index.php?topic=3462.0;allhttps://bitcointalk.org/index.php?topic=27277.0;allhttps://bitcointalk.org/index.php?topic=26278.0;allIf every person on Earth makes ten addresses per second for 20 years (2x1018 total addresses), then the probability that two of these addresses collide is about 1.57x10-12.
A committed individual or organization could easily aquire network storage in the Petabytes. I think that would be more than enough to get a sizable operation started.
1 petabyte is 10
15 bytes.
There are 2
160 possible BTC addresses, each of which is 160 bits == 20 bytes long.
So to store all of them you need 2
160x20 bytes, which is 29,230,032,746,618,058,364,073,696,654,325,660 petabytes.
Bear in mind, we are not looking for a single address among the clouds here. We are looking for -any- address containing BTC.
Suppose each of the 7 billion people in the world has 1000 unspent addresses. On average you would need to try more than 10
35 addresses to find each spendable one. Suppose you can check a million addresses per second, this is going to take you more than 10
21 years.
If everyone in the world is trying to crack this at the same time, it will still take around 10
12 years. And when someone finally cracks it, after paying the electricity bill for 10
12 years, they might be disappointed to find that the key unlocks just 0.05 BTC from the Bitcoin Faucet. Even if it's ten million bitcoins, it's not going to pay the electricity bill for 7 billion computers running for a trillion years.
In short, don't worry about it.