As I know,we can send bitcoin from our own account to other's account without specifying private_key. So I wonder why we have to store our own private key carefully
Private key gives you access to your account and your funds. Public key is used for receiving funds to your private account.
Similar to a bank account, only you can log in with your private information. But anyone can send you money if they know your account number which can be public.
If someone gets your private key, they will thereby have access to your account and take all your funds. Don't ever share your private keys.