Imagine you have a million dollars worth of cars. If you want to store that in a bank, you'd pay them money to do so. Why? Because the car has no value to the bank. The only thing they can do is store it in the vault, which requires security personnel, space, climate control, etc.
Now instead you have a million dollars in cash. In the current environment, where more people want to put money into the bank than take it out, the cash also has no value to the bank. They can't loan it out again because no one wants to borrow that much. So they charge you for storing your money. This is how they make a profit. By slowly taking your balance, since they can't make money loaning it out.
A negative interest rate basically means more people want to save money than spend money. When the government offers negative rates, it's because they want people to spend money instead of save it. When a bank does it, it's because more people are putting money in than taking it out.