There's two really large issues to address here, in my opinion, when it comes to "hate" and where it comes from.
1. Tribalism -- with anything chosen that people take as part of their identity, they tend to become emotionally entangled in it. This can be something as consequential as political party or something as trivial as sports team. When someone chooses to be a part of something, it often becomes an emotional investment. That makes it difficult for people to converse in a neutral manner.
2. Some issues are personal -- It's fine to talk in a dry and academic way about tax proposals or foreign policy. But when the political issues are ones specifically of inherent identity--race, gender, sexuality, etc--it no longer becomes possible to talk in a detached, neutral way, nor should one necessarily. It's not "just a disagreement" when the argument is over whether you deserve rights or protection from abuse. And after a few battles of that level of intensity, it becomes difficult to engage on any of the other political topics with cool civility. After you've argued with someone about whether you should be marginalized, or others should be marginalized, having a cheerful debate about taxes seems a bit silly.
Hatred certainly exists, but generally when people invoke "hate" in this context, what they really mean is a lack of desire to engage with the other person because they view the other person as passively or actively inimical to who they are. It's not hard to understand why, in my opinion, in light of those two things.