I had to look it up. The full context is:

So the new communications strategy for Democrats, now that their polling advantage is collapsing in every single state… collapsing in Ohio. It's collapsing even in Arizona. It is now a race where Blake Masters is in striking distance. Kari Lake is doing very, very well. The new communications strategy is not to do what Bill Clinton used to do, where he would say, "I feel your pain." Instead, it is to say, "You're actually not in pain." So let's just, little, very short clip. Bill Clinton in the 1990s. It was all about empathy and sympathy. I can't stand the word empathy, actually. I think empathy is a made-up, new age term that — it does a lot of damage. But, it is very effective when it comes to politics. Sympathy, I prefer more than empathy. That's a separate topic for a different time.

Later on Twitter:

The same people who lecture you about 'empathy' have none for the soldiers discharged for the jab, the children mutilated by Big Medicine, or the lives devastated by fentanyl pouring over the border. Spare me your fake outrage, your fake science, and your fake moral superiority.

https://www.snopes.com/fact-check/charlie-kirk-empathy-quote/

It's not as bad as the out-of-context quote, but I'm having trouble even wrapping my head around it. I guess the argument is something like:

How can you claim to have empathy when you actively ignore or dismiss the pain of these specific groups? Your empathy is not real; it's a political weapon. Fake outrage, fake science, and fake moral superiority used to win arguments and elections.

He's not wrong about (many) Democrats. But even setting vaccine denialism aside, the core of favoring 'sympathy over empathy' is kind of unavoidable. It feels like tankie whataboutism: 'Democrat's empathy is fake, therefore, more distanced sympathy is our justified approach'