If you don't have a shiny Pokemon yet, you might be eager to get your hands on one. Another thing could be trading the shiny Pokemon for a Pokemon they need for competitive/pokedex (for example, trading a shiny chimecho for something like lugia)
It's fun to collect shinies, and it feels good when you get one. So, there are 2 main reasons why people hunt for shinies:
-Because they like to collect shinies or want a specific shiny Pokemon
-To trade for a Pokemon they want to use to breed for competitive or a Pokemon not yet in their 'dex
Shiny Pokemon are elusive, people may want to find the very rarest Pokemon in the game, or they might want it so they have a good Pokemon to trade. I doubt many would try to capture shinies for bragging rights, since it wouldn't matter anyway and shiny Pokemon are much less rare than they used to be.
In the beginning they probably had the value of an event legendary (shiny chimecho from generation 3 would be extremely rare, since chimecho is very hard to obtain in those games. If it were EV trained with good moves, then it would be even more rare and one of the most valuable Pokemon), depending on the Pokemon.