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A myth. The skin shrinks back. The nails just look longer.
Where it came from
People who exhumed bodies or sat with the dead noticed what looked like longer nails and stubble. It is a striking sight, and it found its way into folklore, vampire tales and, later, novels.
What’s actually true
Growing hair and nails needs cells to divide, and cells need glucose and oxygen, which stop arriving when the heart stops. After death the skin dries out and retracts, pulling back from the nails and the base of the hairs and exposing more of them.
Morticians moisturise the skin of the dead partly to prevent this effect.
Further reading
- Wikipedia: List of common misconceptions (human body)