A lot of the responses are correct, but there is one aspect being missed.
Liberals don't NEED to hate conservatives. There are real problems in the world that the left is trying to remedy.
Conservatives NEED to hate the left. Modern conservatism (and some would argue all conservatism) doesn't have any moral ideology. There is nothing they're fighting to for. Conservative ideology is the idea that there is a group that the law should protect but not bind and a group that the law should bind but not protect. To push this, an out-group has to be created and hate is the only way to dehumanize someone enough to treat them the way conservatives treat women, minorities, LGBTQ+ etc...
Conservatives hate Liberals because conservatism doesn't work without hate. They hate because they NEED to.