You can't sum up why console X beat console Y with a single phrase!
It was a whole list of reasons, not all of them Sony's doing. Sega made some big mistakes with the saturn and Sony just seemed to take a lesson from all of them.
Sony learnt a lot and managed to market the PS to a different target audience and had a lot more money to go for exclusive deals such as tomb raider.
Electronic Arts also played a major role in the success of the playstation. Without them supporting the Saturn they lost a lot of sales to the Playstation.
The ironic thing is that Sega helped bring the older audience to gaming with the Megadrive/Genesis, and Sony stole it right from them.