Many of them have houses that they sell as my Mom did. If they don't and they have less than $11,000 in TOTAL assets, they can be subsidized by Medicare.
And like in my mom's case, the house that she sold for a nice profit financed her stay in the nursing home. Every month, her private assets were chipped away another $5K. Drip, drip, drip. Gone, or almost gone.
We were really lucky, though. She had money to pay for her stay, and when we (the family) had our ducks in a row, we were able to bring her home to my sisters' home where she was taken care of, primarily, by this sister, but by all of us sisters, until the day that she died.
With the rules being what they are, my mom's life savings was taken to pay for her stay in the nursing home, and for her resulting care.