Say someone sends me 0.5 of a BTC to my desktop wallet, is it a digital file or lines of code that they send me which has monetary value? I'm very confused on what you actually RECEIVE with cryptocurrency.
Think of Bitcoin like a ledger. When you (your Bitcoin wallet) receives Bitcoin, instead of receiving an item or file, everyone is updated with an updated ledger including the balance of your Bitcoin address.
Here is an extraordinary explanation on Bitcoin.
"you receive a bitcoin" means that someone makes a transaction with your BTC-adress as aim. Nothing of this happens on your computer or in your wallet. It's everything in the decentralized blockchain.
Your wallet is only an easy tool to save you private key, to ask the network to make a transaction, to see your total amound... And with the private key, you have access to the adress where someone sent the bitcoin to.