I get what the man did was bad. But just don't know how they caught him by using a new made up at the time science technique.

id heard his brother turned him in

They published his writings and his brother-in-law recognized it. The mini series with Paul Bellamy is definitely worth watching

Linda Patrik first identified the Una Bomber: https://www.thetedkarchive.com/library/a-conversation-with-linda-e-patrik

That's what I drew for reference. Cause toward the end he argues about the new method.

I don't even know if I'd call it a science, just a series of inferential techniques. David Kaczynski believed that the author of the manifesto sounded a lot like his brother, and told the authorities, who analyzed it and debated whether Ted Kaczynski's letters were similar enough to the Unabomber manifesto to be the same person. It doesn't even need to be a robust scientific model to have the conversation. Really no different from sifting through witness descriptions of a suspect's appearance and seeing who those descriptions match. Investigators can compare their experiences/beliefs about which traits are rare or unique and which threads are worth investigating further. In the end, they had enough for a warrant, and the search itself revealed a lot more hard evidence that Ted Kaczynski was the Unabomber.