It's not an idiom.
"To accumulate" means to gather or to amass.
A debt is an amount of money that you owe to a person or company. For example, if you spend £100 on a credit card, you then have a debt of £100, which you will have to pay the credit card company at a later date.
So someone who has accumulated debts will, over a period of time, have amassed a series of debts that they now owe to a variety of different people or companies. Examples would include owing money on your electricity bill, having student loans, owing money on a credit card, having a mortgage on your house, or paying for a large item such as a car over a period of months or years.
Not a teacher.
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