In order to put your father on Medicaid, from which I assume he was in a skilled nursing facility of some type which took and his monthly social security payments and used Medicaid to pay for his care, the system took a lien on his property, such as a house he owned. They don't require that you sell all your assets before you get medicaid, because of course many things like houses don't sell on demand, and there may be others, like a spouse or family member living in the house. But when the person dies, there is that lien, there is that requirement that yes, he did receive Medicaid and yes, he did or his representative did to receive this medicaid, agree to use any assets in his estate to pay it.
The term they use is "pauperize himself." That's what he or his representative had to do to get medicaid. That means someone else has the right to any property or assets he dies in posession of.
In our state, you cannot settle an estate without a document from the state's medicaid program saying that no money is owed to them.
I think you are misunderstanding when they say you owe it. Who owes it is your father's estate.