K so Shiny Pokémon were probably introduced to take advantage of the Game Boy Color which was back then very recent and a miracle of technology. In Gen II, Shiny Pokemon did in fact have a slight (VERY SLIGHT) advantage over their standard variants, since they were shiny based on their IV's - and generally they had slightly better IVs than the standard Pokemon. Only slightly though, but it still gave a small edge.
So back in those days, you could probably understand why people sought out shinies.
So I'm going to extend a little bit on all the other answers which are remarkably similar, and completely true. Shiny Pokemon are great for bragging rights, and were even better back in the previous gens where the chance was a measly 1/8192.
Human Nature, unfortunately, means that a lot of us will tend to hoard things that are rare, shiny and valuable. Like money. o3o. Shiny Pokémon are rare, so there will be people trying to hoard them. And since hoarders want more of these rare valuable Pokemon, they'll be willing to trade strong Pokemon over for random crap like Shiny Bidoof. Okay maybe not to that extent, but you get the gist. It also gives them something to waste their already wasted lives on. Also don't forget the random vain people out there who care about appearance so much - they might care about the appearance of stuff in games too. Yes there are people crazy like that and no they do not have brains, and if they did it would be the size of a peanut.
Additionally, the misconception that Shiny Pokémon have stat advantages still exists over a large portion of the Pokémon Community, and as such that is another reason.
So lets assume for a moment, the people who know that Shiny Pokémon have no stat advantage whatsoever over the standard forms are playing Gen 6, which has nearly doubled the Shiny Chance. Why would so many people still be going for Shiny Pokemon? The factors of bragging rights still exist, and also since a lot of people have rarely ever encountered a shiny Pokémon, due to the extremely low chance - and now with a higher chance, they're taking advantage of it to get more shiny Pokemon and fulfilling their innate desire for all things bright and sparkly.
TL:DR? Human nature sucks, people like sparkly things, and people still think shinies have special properties.