how does the blockchain.com/explorer page know if an address is included in a wallet?
Short answer: it doesn't.
Blockchain.com just looks to see if it has a record of a transaction involving the address you give it.
But you don't need to use bitaddress.org or any similar service and I wouldn't.
A little more detail.
The web-page at Bitaddress.org uses a pseudo-random number generator based on mouse movements and keyboard input to generate a random private-key.
Form a private-key you can derive a public-key. The bitcoin-address is a hash of the public-key.
The private-key enables you to spend bitcoin and should never be shared with anyone else.
The public-key enables other people to send you bitcoin, you give them the address derived from the public key.
There need be no link between bitaddress.org and any blockchain explorer. The latter just searches the blockchain for transactions involving the address you type in.
The blockchain is a ledger of all bitcoin transactions from the creation of Bitcoin 10 years ago. All "full-node" wallets have their own copy of the whole blockchain.
Despite the above, I would not use a public service like bitaddress.org to create my private-key. After all, we don't know if the owner of that website dishonestly keeps the private-keys (or has been hacked). Bitcoin was originally designed to be used without trusting third-party services like this. You can download a desktop-wallet and generate your private-key in the privacy of your own home on a computer that is disconnected from the Internet. That is the one of the safest ways to inexpensively create a Bitcoin wallet.