I'll put it down to several points, but basically I feel like the PS1 was the perfect storm in several ways.
1) Great marketing. Sony basically took the kind of advertising that Sega pioneered with the Genesis and made them even cooler. Meanwhile, with Saturn, Sega had bizarre ads that appeared to target some kind of slacker/yuppie hybrid audience which felt uncool as fuck.
2) Great aesthetic feel to the product: The console had a nice flat aesthetic that made it feel more like a audio deckbed rather than a piece of gaming equipment. Even the boot up sequence oozed coolness.
3) Technology felt solid: It seemed to *just* work for 3D, crisp, fast and good looking. By comparison, the Saturn's 3D was slower, the visuals were less defined, even things like translucency looked wrong with mesh patterns and such. PS1's FMV looked great too, while Saturn's looked pretty bad. Even memory cards felt like good technology: small, no batteries so no risk of losing saves, compared with Saturn's internal memory which was prone to running out of battery power, or the save cartridge which was a massive block thing. 2D was certainly inferior to Saturn but for most consumers, the 3D the PS1 did was "good enough". Also the PS1's cheaper price, even though it appeared to do much better 3D than the Saturn really impressed people.
4) The competition happened to shoot themselves in the foot: Sega had their very expensive, complicated, underpowered (3D) monstrosity, while Nintendo remained with outdated cartridge technology and further tightened the screws on 3rd party. The 3DO, Jaguar, etc were all DOA. Sony virtually won the generation by default.