What “not yours” means
When you buy crypto on an exchange, they write in their own ledger that they owe you that amount. The crypto itself is still at their addresses. While the exchange is working, the difference is not felt. When the exchange has a problem, it is felt entirely.
This is not theory — several large exchanges have collapsed with customers’ money in the history of crypto. Hence the saying that crypto not in your wallet is not yours.
When an exchange is fine
If you are buying a small amount and plan to sell it within a few months, keeping it on the exchange is practical. Withdrawing costs a network fee, and for small amounts that fee can be disproportionately large.
If you trade actively, the money has to be there anyway.
When a wallet is worth it
When the amount is one it would hurt to lose, and when you do not plan to touch it for months. That is when it makes sense to move it to a wallet whose key only you hold.
There are apps on a phone and dedicated devices shaped like a USB stick. The device is safer because the key never touches the internet.
The responsibility that comes with it
Your own wallet comes with full responsibility. Lose the seed phrase — the twelve or twenty-four words that restore the wallet — and the money is gone for good. There is no support desk that can bring it back.
No legitimate party will ever ask for your seed phrase. Not an exchange, not support, not a website. Anyone who asks is trying to rob you.
The first time you withdraw, send a small amount and check that it arrived before sending the rest. A transfer cannot be reversed.