It was a combination of the processing power of the supercomputer IBM built specifically to run it, and the sophistication of the software engine they designed to execute that power. They had teams of computer scientists and grandmasters working together to develop this. This code was a tighly guarded secret, and I've never looked into whether or not the code was opened to the public. It was also designed specifically to defeat one opponent, Garry Kasparov, so there is the possibility that extra care was paid to openings they believed him most likely to play. Apparently they were tweaking the code even during the matches themselves, figuring out where the machine failed, and trying to prevent that failure in future matches.
Kasparov famously believed the IBM team cheated, and used grandmasters to help feed the machine moves at critical positions. There was a documentary that investigated this theory (with a rather heavy pro-Kasparov bias).
https://en.wikipedia.org/wiki/Game_Over:_Kasparov_and_the_Machine
https://www.imdb.com/title/tt0379296/
The competition with chess engines is to write code that is the most efficient at analyzing positions intelligently. The reason is there are far, far more positions than any computer-- even today's supercomputers-- could hope to analyze. We're talking about millions of years. So you can't brute force it. Essentially, what a chess engine is supposed to do is dispense with calculations that are the most likely to waste time by assessing terrible moves. And, in this sense, they're not that much different than the human mind. The tricky part is figuring out how to teach the engine which moves are most likely to be bad, especially since we're learning from the engines, now, that certain moves we wouldn't have considered can be viable-- at least if played by a computer not prone to human mistakes-- are actually the strongest in a given position.
You can actually play some of those old chess engines against the best open-source engines today. Of course, today, Stockfish is king. Plug Stockfish into an old game like Chessmaster, and Stockfish will kick the shit out of Chessmaster on its most competitive setting. This is running on the same hardware. The software is vastly superior. So this is an ongoing competition in the chess world. The golden standard of these competitions is the one that controls the hardware. In other words, which engine outperforms the others on identical hardware? There is a Twitch channel that specializes in showing nothing but games between chess engines:
https://m.twitch.tv/computerchess/clip/CoweringFineCrabMrDestructoid