What happens to your property if you die in a nursing home. does the nursing home take part of the value of the home?

A nursing home has no more right to your property (in general) than any other creditor has. If you die owing money to the nursing home, then they're entitled (as is every other creditor) to recover that money from your estate. Otherwise, no, they don't get to take part of your property unless you signed a specific contract with them stating so.