> The buyer takes out a loan, how does that become the responsibility of the company they purchased?

Because the company they purchased is now a part of them.

As for why a lender would agree to it, it's because these transactions are not as simplistic or universally disastrous as they are usually described. A lender will obviously only make that loan if it has a reasonable expectation of being paid back, and most of them are. They may get additional collateral like parent/affiliate guarantees and the loans will have covenants relating to financial performance etc.