Not Your Keys, Not Your Coins: What Does It Mean?

Not your keys, not your coins means that if somebody else holds the secret key to your crypto, the crypto is not really yours: you own a promise from that company, not the coins. When you buy bitcoin on an exchange, the coins sit in the exchange's own wallets and your balance is a number in its database. It usually works fine. It stops working the moment the company does.
Why does the key matter so much?
On a blockchain, ownership is not a name on an account. It is the ability to sign a transaction with a private key. Whoever can sign can move the coins, and nobody else can. Holding crypto on an exchange is like leaving your coat in a cloakroom: you have a ticket, and the ticket works as long as the cloakroom is still standing and still honest. Hold the key yourself, in a wallet such as MetaMask or on a Ledger device, and there is no cloakroom at all.
Has this actually gone wrong?
Repeatedly. Mt. Gox collapsed in 2014 owing customers hundreds of thousands of bitcoin, and creditors were still waiting more than a decade later. In 2022 the lender Celsius froze withdrawals and the exchange FTX failed, and in both cases people who could see a balance on screen discovered they could not take it out. None of those users had done anything careless. They just did not hold the keys.
So should everything come off exchanges?
Not necessarily, because self-custody moves the risk onto you. There is no password reset in crypto: lose your seed phrase and the coins are gone for good. Plenty of people have lost more to a mislaid backup than they ever would have to an exchange failure. A reasonable middle path is to keep on an exchange only what you are actively trading, and move long-term holdings to a wallet you control.
If you take one step this week, make it this one: withdraw a small amount to a wallet you control, write the seed phrase on paper, store it somewhere only you can reach, and send a test transaction back. Do that once while the stakes are low, and the decision stops being theoretical.