The best way to understand a loan is as the right to a future income stream (principal repayments and interest). The original debtor (the person/entity taking out the loan) establishes the credibility of that future income stream (based on income, expected returns on a project, etc) and sells it to the lender (usually a bank) for cash up front. Thus the loan is an asset on the bank's balance sheet, that is generating returns (assuming all goes to plan). Banks can and usually do sell on that asset to other parties.

Conversely, when you deposit cash to a bank, you are actually creating a liability on the bank's balance sheet - as you might want your money back one day!

For every debt there's a debtor and a debtee. For the first debt is a liability, while it's an asset for the second.