How on earth did the Unabomber evade capture for nearly 20 years, despite the costliest FBI investigation in history?

So my specialty is in militant left wing terrorism, but Ted K has been a fascination of mine for years because of his manifesto and the response it garnered, as well as the effects his actions had on domestic bombings in general.

The short answer is sort of what you'd expect for any criminal getting away with something for a long period of time: he was absolutely meticulous in his planning and preparation.

The long answer has to take a lot of smaller but equally important details into account. One thing that is incredibly interesting to note: many of the investigators who led this case remark on how Ted K's bombs became more and more sophisticated over time. The first bombs were triggered by common match heads. But it's not the deadliness of the bombs which seriously affects the investigation's efficacy: it's how professional he became at concealing any genetic traces/pieces of evidence. Keep in mind that the preliminary bombings were NOT subject to an FBI investigation, as they were not yet considered federal crimes. Instead, they were investigated by the Postal Service itself which - despite it sounding sort of strange - has a very professional investigative and policing unit. Still, it didn't have the bankroll or clout of something like the later UNABOMB investigation. I will not speculate on whether or not Ted K made mistakes with his earlier bombs which could have identified him, but it pays to remember that the famous FBI investigation everyone references actually starts in 1979 after the attempt to bomb AA Flight 444.

Now, on to the more boring part. The investigators tasked with finding the UNABOMBer remarked often that they could not find any hard traces which could identify the bomber. Terry Turchie, who headed the task force, made this point pretty clear. James R Fitzgerald - whose story the Discovery series "Manhunt" is at best LOOSELY based on - remarked that Ted K would literally burn down commonly bought items like batteries to produce the materials he needed, and fashioned trigger devices from wood he crafted himself, making him untraceable.

What was left to catch him? Well, forensic evidence left over during construction: hairs, finger prints, etc. He was meticulous, by his own accounts in the collected writings he put out from prison after arrest, at eliminating these issues. He hand sanded the wooden boxes the bombs were packaged in. Hairs were removed immediately.

And this is to say nothing of the fact that, as you stated, the investigation spanned decades. Forensic science in 1979 was very different from even 1989. Camera surveillance of post offices was not as common during the period of the bombings, as both the UNABOMB task force and the Postal Service investigators have remarked after. Not that it necessarily would have helped, given that Ted hand delivered some of the bombs, or used fake return addresses which would have the bombs double back to a target anyway.

The last point is one I think gets overlooked: he went dormant. For quite a long time, as a matter of fact: six whole years passed with no incident. This does not mean the investigation stopped, but it does mean that new evidence stopped coming in and - given his proclivity for preventing forensic evidence from contaminating his devices - made the investigation much more difficult. I have read in Fitzgerald's books "A Journey to the Center of the Mind" that there was also departmental infighting about motive, as the bombings started during the late '70s when left wing radicals were a more likely perpetrator than a neo-luddite loner. This last point I have only ever found in his memoir, so I still take it with a grain of salt, especially given how that Discovery series oversold it.

As for his capture, I'd be happy to elaborate on what I know about that.

For sources, I highly recommend "Industrial Society and its Future", which will give you an insight into the mind of the man who perpetrated these acts. His "Technological Slavery: Collected Writings" have a couple of essays that deal more with the pragmatic steps he took towards his bombings. All three segments of Fitzgerald's "A Journey to the Center of the Mind" are enlightening, though the last book is the most relevant. Turchie and Fitzgerald both spoke to NPR in 2017 about the issue after the Austin bombings, and their comments support the idea that his meticulous forensic evasion was important.

EDIT: Sorry, I forgot the source which was probably most important but it's not a google link away: UMich has his papers and several case files which were provided by the FBI in a special collection. If you are a visiting researcher with even a modicum of interest, it's incredibly cool. I've never reached out to ask about online access.

SECOND EDIT: Sorry, I promise no more substantial edits outside cleaning up my garbage prose. Here is a news release from 1995 listing some of the top suspects. It is a hearsay article, though one whose Top 5 is pretty similar to what the case files implied. You'll notice the people of interest have particular political motivations in common. In short, even up to 1995, the wider investigation had the analysis of UNABOMB wrong.