According to Oxford Languages,
sign (a check or bill of exchange) on the back to make it payable to someone other than the stated payee or to accept responsibility for paying it.
I'm having trouble understanding the bolded part. "sign (a check or bill of exchange) on the back to make it payable to someone other than the stated payee" is clear. This basically means that if my father is the stated payee on the cheque, and he endorses the cheque using my name, then I will be able to encash the cheque as well. However, what does "to accept responsibility for paying it" mean?