Multiple factors including:

• Cheaper hardware- Sega Saturn and N64 were both more expensive than the PS1. With N64, the higher priced games didn't help
• Large variety of games from a variety of genres.
• Advent of 3D meant many genres were created or popularised on PS1. This was new and novel at the time.
• Large number of exclusives. Easily the most important point. Many franchises were started on PS1 as well as more established franchises moving into 3D:
  
  • Tekken,
  • Resident Evil,
  • Metal Gear Solid,
  • Final Fantasy VII-IX,
  • Wipeout (technically not exclusive but the studio was bought by Sony),
  • Syphon Filter,
  • Crash Bandicoot,
  • Ace Combat,
  • Dino Crisis,
  • Silent Hill,
  • Tony Hawk,
  • Vagrant Story, 
  • Gran Turismo,
  • Time Crisis
  • Ridge Racer.
• Some multiplats were also very popular on PS1 like Tomb Raider and didn't have an equivalent on N64. Even the wrestling games like Smackdown were great on PS1.
• Some games gained notoriety for being violent. Some like Thrill Kill were also banned but available online. 
• More mature focus- it became more acceptable for young adults to own a console meaning a larger market to purchase the console on top of the younger generations. 
• Linked with the top point, the games were a lot cheaper than the N64 carts
• Easy to pirate meant lots of hardware sold.

Third parties, CD format, third parties, third parties, and third parties.  The PS1 was the console where third parties really cut loose.  Sony provided Eidos, Capcom, Konami, Square, etc. a console that they could go crazy with.  And go crazy, they did.  Boundaries were pushed, new ideas were realized, and Sony (and gamers) reaped the benefits.  Final Fantasy 7, Metal Gear Solid, Resident Evil, etc. these were games that Nintendo's 64 bit console was just incapable of producing without some serious technical limitations (I know about RE2 for the N64.  Shut up.).

Sony didn't really do anything special.  Nintendo cut their own balls off by sticking with the cartridge format and Sega screwed up by creating a console with the word "Sega" on it (gamers were pissed with Sega by the time the Saturn launched.).  With no place else to turn, the fairly solid PS1 was the heir to the throne.  If you were a gamer without a PS1 from 1996-2000, I really pitied you.

for me the games of course, crash bandicoot1-3(best platformers ever yh, even better than mario ),FF7-9 (some of the best in the series), tomb raider,and many many MANY others made the ps1 the best of its generation, and of course the controller, n64 is so uncomfortable a total piece of garbage. nintendo should fire the asshole that desing that shit

Spyro and Crash Banicoot and Ape Escape were my childhoods most favourite games 

It had everything and then some and felt fresh and unique.

The advantage of N64 is multiplayer. For single player, PS1 have literally dozens and dozens more great titles whatever genre your in. By that time, I guess you were mostly playing couch multiplayer. If not, you would have gained from being more open minded.

This.  PS1 and SNES were the best consoles for the genre, IMO.

Even if you go all time, they are probably two of the top contenders for best ever along with PS2.  Not going by importance, but by games.

Variety of games and N64 was barely available where I lived so I didnt know what it was until I saw it at America. But eah variety made it what it was. Kinda miss those days.

Great RPGs is what makes it special