Crypto staking is the process of locking up crypto holdings in order to obtain rewards or earn interest. Cryptocurrencies are built with blockchain technology, in which crypto transactions are verified, and the resulting data is stored on the blockchain.
Your coins are still in your possession when you stake them. You're essentially putting those staked coins to work, and you're free to unstake them later if you want to trade them. The unstaking process may not be immediate; with some cryptocurrencies, you're required to stake coins for a minimum amount of time.
In most of the cases, the type of coin and where you stake matters a lot and can determine if your staked coin is safe because if the platform could be hacked and access is gained to the vault where coins are stored it could mean a lot. For example Quidax has a tight security system that stored crypto offline (cold storage) and so the QDX token that's staked by customers are safer than any other exchanges storing coins online (hot storage).