When you deposit a sum of money into a bank account, that account is an asset for you. At the same time, it is a liability for the bank.
Let's say that you are depositing $1000 to your bank account. You no longer have that cash; the bank does. However, that account is an asset for you worth $1000. At the same time, the bank now has that $1000, which they can then use for other purposes. They have a $1000 liability to you, which means that they owe you $1000.
As you mentioned, per your agreement with the bank, they may pay you interest and/or provide services to you while they owe you that money.
It is very similar to you taking out a loan from the bank. In that case, the bank is the one that gives you cash. You can now do what you want with that money. They have an asset (the note), you have a liability (how much you owe), and per your agreement, you owe them interest while you hold their money.