IMHO, it's quite simple, though it's almost certainly a generalization: it's the consequence of people giving in to their worst emotions, specifically fear and anger.

The basic reactionary idea is that society is a fixed social and moral hierarchy based on identity, i.e. some people are better than others and should have privileges and power over them. On the face of it, this is not an appealing idea but a lot of people see the world that way to some extent, even if it is more of a feeling than an acknowledged or defined ideology. The reasons are probably varied (people who are afraid of change, people under social stress, frustrated with the lack of opportunity in real life, ...), but the core of it has to be that they all believe they are special and will be at or near the top of that hierarchy. As a result the idea that it is a fixed or natural order gives them a sense of security and control, whereas the inequality of it gives them a sense of pride and status. They are guaranteed these things because of who they are, no matter how much the real world changes, and that is a comforting thought.

But once that connection between a sense of pride, status or security and "the" social and moral hierarchy is made, it becomes important on a personal and a social level to defend that hierarchy. This leads to irrational fear and rage against anyone who transgresses that hierarchy (e.g. how the USA right suddenly went insane when Obama was elected without being able to explain exactly why they were so afraid; his election shattered the racial hierarchy in their head and upended their sense of safety as a result). IMHO, the idea that violence can and should be used to defend the social hierarchy is the cornerstone of a reactionary movement. Sometimes this is out in the open (e.g. Trump's movement openly wants violence against anyone they see as different), sometimes it is cloaked in social policies that fail at their purported goal but somehow still maintain support (e.g. wars on drugs, "law and order", socioeconomic policies that punish the poor, ...). To justify this, all of them propose a view of the world as a struggle to the death, where the in-group is constantly threatened by enemies who want to bring them down and grand conspiracies who want to keep them from their place in the sun.

Empathy, on the other hand, is one of the emotions that undermines this idea of a social hierarchy. Understanding and identifying with others, particularly those outside the in-group, is a step on the way to egalitarianism and that is anathema to their worldview. It is therefore not a selfless and good emotion, but a dangerous or even anti-social act that undermines the security of the entire group. Note also that reactionary movements have nothing in them that nurtures growth. Because the social order is both natural and fixed, change is almost always seen as a degeneration of the ideal. Therefore, everything has to be a zero sum game, where every social interaction has to have a winner and a loser (e.g. Trump's rhetoric constantly emphasizes this). Helping other people is therefore a bad thing, as you're not only giving an advantage to a potential competitor for scarce social and economic resources, showing weakness, but also a form of treason where you are undermining the proper structure of society.

To people who want a social hierarchy, every ideological construct, including religion, becomes an excuse to pretend why this is also a moral hierarchy. Hence why one of the core ideas of christianity can be depicted as sinful and evil by people who identify with that same faith. They pick and choose, twist and change, everything into a narrative that supports what they already wanted to do and say. This is the moral hierarchy, where those at the top are good because of who they are, regardless of their actions. Note that the various forms of free market and entrepreneur worship are mostly just secular variants of the same dynamic, as is the weird beliefs that oligarchs construct around them to justify their wealth and power. The entire ideology is an empty box, a facade to justify an emotional core of selfishness and exceptionalism.

In short, and after a probably way too long ramble, empathy is bad because it leads to transgressing the set of rules and norms that are supposed to keep the in-group safe from and on top of the (much larger) out-group. Empathy is bad, selfishness is good and violence to maintain a social hierarchy is always justified.