What are skins?

"Champion skin (or just skin) refers to the color scheme or appearance of a champion."

(Taken from the Champion Skin page) 

We probably all know what skins are. They're cool (most of the time) because they basically change the appearance of the champion you're  playing as. Some skins are worth the RP. Some skins are better off not being purchased or are for the sake of collectors. There are skins that just change the appearance. Some have particle changes. Then there are the ultimate and legendary skins. They all serve the same purpose. But my question is: What, exactly, is the purpose of  these skins?

Something that happened to me a while back

Here's a little story about one game I played. It was an ordinary Blind pick game. I wasn't able to enter ranked queues just yet so that was the only choice I had. I was playing as mid-lane Viktor (seriously, though. Why do few people play him nowadays?) and the enemy team had a Katarina for their mid-lane. Katarina has got to be my least favorite champions because of how powerful she can be when fed. And to add icing to the cake, their Katarina had a skin (I forgot which one)  Normally, I would think that if you used skins on a champion, you're really good at using that champion. You'd think they're pros at the champions they are using. 

But I was mistaken. That Katarina was an idiot and a feeder. I got fed in lane and carried the rest of my team to victory. My team was taunting the enemy team with mostly "Your Katarina is stupid. The skin's a waste."

Another thing that happened to me

This was another game I played a while back. Played mid-lane Swain. The enemy team had a top-lane Lee Sin who had the Muay Thai skin (That skin is OP, btw) I pretty much predicted that their Lee Sin would be good. And guess what? He was. He and I had pretty much the same score, but my team just kept dying to him and the rest of their team, which meant I lost that match. 

The Question

What are the purpose of skins? Are they just for show and that you're one of the lucky one who has RP to spend? Or are they indicators that you're dealing with expert players that will pretty much own the entire game? Or maybe you got it from a mystery gift and it'd be such a waste not to use it? I'd like to know your thoughts and opinions on this.