Nintendo dropping the ball with 3rd parties left a huge void that Sony exploited. Sony also had more lax censoring policies which lead to a more mature perception (although the same could be said of SEGA.) They were in the right place at the right time.
Really though, it was ALL about third parties finding a home, no one played the PS1 for Sony 1st party games.... hell, the only ones I can remember owning (off the top of my head) are Parrappa (which I love), Twisted Metal 1 & 2 (also awesome), and Legend of Dragoon (which was only ok in my opinion.)
Mean while, Konami (Metal Gear, Castlevania: Symphony of the Night, Suikoden), Sqauresoft (FF, Chrono Cross, and so many other JRPGS), Eidos (Tomb Raider, Legacy of Kain at the time, right?) and others provided all the really memorable Playstation experiences. It is one of the reasons Playstation All Stars wasn't a hit.... PS Allstars are largely 3rd party.







