'unsolicited' means 'not asked for'. So - the credit card companies were sending these people credit cards in the mail. The customers had not asked for the credit cards. 'large borrowing balances' I assume means 'large credit limits'. That means that they could put a lot of purchases on the credit card.

Unsolicited means the customer never asked for it. A "large borrowing balance" is a strange way of referring to a generous line of credit. Before the last financial crisis, banks used to send credit cards to young college graduates who hadn't requested the cards, knowing that such graduates are a good credit risk. The young people tended to get a little carried away with spending. Banks discontinued the practice after they became more risk averse.