What is a crypto wallet?
A crypto wallet does not store your coins. Your balance is recorded on the blockchain, a public ledger anyone can inspect. What the wallet actually holds are the keys that prove that balance is yours, which is why people say you are not really moving coins around, you are proving you have the right to spend them.
Two pieces do the work. Your address is public. Your private key is secret, it signs transactions and authorises anything leaving the wallet. Whoever holds the private key controls the funds. That single sentence explains most of what follows.