He has insane mobility, negligible energy costs essentially removing all ability constraints aside from cooldowns, a passive restoring energy and giving him an AS steroid (Synergizing with high AD values), massive AD scaling, % missing health damage, and very large base damage values. He can build tanky and still deal respectable damage, and scales well with relatively small sums of AD due to said ratios. He has high mobility, and good survivability. So it's established, his raw effectiveness in combat is very high.
But wait, there's more! He has a huge quantity of utility. Revealing stealth, one of the largest displacements in the game, a shield, a movespeed slow, the ability to check bushes via Q, lifesteal/spellvamp innately increasing survivability, the sheer mobility allows insane playmaking with wardjump, and it causes the CC/damage/utility/mobility to synergize entirely too well.
But it doesn't stop there. His early game is insane. One of the best. His ganks are just absurd because he packs CC lockdown with damage. On top of this, it's hard to win 2v2s because of it should he be counterganked. This, plus the fact he is strong early means he's impossible to counterpick, making him a safe pick all of the time.
The fact he has such an oppressive early game means he should fall off. Basic design says that, power curves that are intense early, are typically mild later on, correct? Hahahah, of course it i-WRONG. Lee Sin is strong earlygame and can get himself fed pretty consistently. He scales well with the gold, due to his scaling and playmaking design, meaning even his power curve synergizes with his kit, and is NEVER actually weak.
End point, usually with EVERY SINGLE champion, I can point to something and say, "That's bad. It's an exploitable weakness that can be used against him/her" but with Lee Sin, he just isn't built with any weaknesses, and as much as Lee Sin players hate to admit it, he doesn't really fall off.
TL;DR: He's just oppressive and is never a bad pick. Even if he does poorly, he's still useful. He's never actually weak.