How is Fire good against Water? Well put some water on a fire and its gone.
What? That doesn't make sense XD
How is Fairy good against Dragons? Old mythologies told stories about Fairies killing evil dragons.
I have never read a myth like that. I've read over thousands of Dragon stories and myths, majority of them are real world equivalents with humans outsmarting dragons. Fantasy ones dragons are always immune to Magical attacks and Fairies (good and Evil) feared Dragons. The only time Magic could effect a Dragon is through mediums, Enchanted wapons or Magical chambers to trap dragons by enforcing the structure or making it immune to a Dragon's attempts to escape. Dragons themselves were never effected directly by Magic. In modern media, Dragons have some of the highest magical resistance to every type of Magic except special Dragon Slaying Arts which is a combination of Magic infused Martial arts.
How is Psychic good against Poison? If you're strong mentally you can overcome the most dangerous and deadly fluids.
This is not even close to being true. Being strong mentally will not save you from a cobra bite nor will it save you from severe poisonings from other substances. No amount of mental power will keep you alive when poisoned, only medication and proper medical treatment can save you.
There are many real world applications that don't make it to Pokemon games. Cold can overcome heat. More cold than heat can kill objects as well as pure cold can kill a fire faster than it normally burns. Flying types being weak to rock doesn't make any sense. Try throwing rocks at birds to see how many you can hit. Unless you have something that gives off a lot of speed and propulsion, you won't hit your mark. Though if you have something with enough speed and propulsion, metal can kill birds, plastic can kill birds, hell, wind can kill birds so why aren't Steel and Flying super effective against Flying type?
Too much water can kill plants, why can't Water be neutral or Super Effective to grass?
Electricity may pass through water, but it only effects objects within the water. If the water has no particles (Pure water) for the electricity to pass through then the electricity won't effect water. Why is Electric SE against water when the purest form isn't effected by it IRL?
As Archeops said. Its Pokemon Logic. It doesn't have to make sense to be applied. The company can say anything they want, make up anything they want and pass it off in their games.
I do not like Fairies because of the reason they exist. Don't like the design of a few of the new ones and hope there are some more menacing ones to come in the near future.