They hate her because she is a true beliver is His word and not some phony run of the mill RINO politician who is either ashamed of Him or uses Him as just another prop to advance their political career.
This is what they truly fear and they know that there are plenty of Americans (i.e., voters) who believe the same way.
I believe the greatest hatred is comes because of the guilt that racks their souls over abortion. She saved and loved one of God’s innocents...and it is plain to see the happiness that child brings the Palins and they to the child. They advocate killing those precious babies...and many of them have done so and the guilt racks their them over it...so they lash out at Sarah.