First of all please don't add unnecessary "size" to your comments it is annoying. Instead change the "zoom" level of your browser if you have trouble reading.

I'll try to explain:
What you need to know about bitcoin is that there is no such thing as "bitcoin" and you are not "storing coins" what you have is a private key which creates a public key, and record of all transactions that are stored on this public ledger called blockchain that peers hold on their computer when running a full node.

When you receive bitcoin, what happens is that some else is signing away the ownership of an output to your public key. And when you want to spend that output you sign away ownership of that output with your private key and give the ownership to another public key and all the records of these transactions between people are saved on the blockchain so your "database" is the blockchain.

What you do offline (generating a paper wallet) is that you are creating that private key to be able to receive those outputs and be able to spend them later.

For checking your paper wallet's balance, you go to that database (blockchain) and check your public key and see how many outputs were sent to your pubkey and add them up to see how much balance you've got. You do this usually by going to an online block explorer and typing in your bitcoin address (base58encrypted) starting with 1 or 3.