Corki is popular because:
He is an ADC that does magic damage. So, even in a full AD comp, Corki is bringing you mixed damage.
He builds AD, so his damage on turrets as well as minions (early game farming) is above that of APCs.
He is quite safe in lane.
And then, every 5 minutes. He can destroy the enemy laners.
Like. You lane for 8 minutes, get your lv 6+. You get your Sheen. You go pick package.
And then you fly to one of your lanes (where you don't belong).
I prefer doing this to the bot lane, as they are under leveled, and I can get 2 kills over 1 kill on top/mid.
So, you come in, you use your W from the fog of war. You knock aside the enemy bot laners, zoning them out from their escape path.
While they try to realize what happens, you QE them and spam aas and missiles.
Why this works?
You gank from lane, there's no lane wards (if there are any wards, they are on the river). So, the enemy won't see you coming.
Package range is huge.
You got a 40% movement speed buff while running with Package in your "inventory". So, you get to bot/top/mid (in case you are top laner) 40% faster than normal.
And the damage this thing does if the enemy tries to ignore it and go though that path is huge also a huge slow. So, you can use it in such a way that the only way for your enemy to fight is moving towards your team. Which usually means suicide (especially if the enemy adc has to walk towards your adc and your support).
Since you get your W off cooldown, even if they flash away, you can W after them to follow up.