Playing other champions really gives you a feel for how to handle situations. It is important to know what your champion is capapable of and what they are capable of. I can offer some tips though.
Watch your map: This one is the most important, always watch your map. When more than one person is missing on the enemy team you should be moving to a safe position. Make sure you know where your enemies are or where they were heading last seen. Keep the map scouted and when 3+ people are missing you should be trying to bunch up aswell. If theres 3 or more people missing and you're pushing a lane deep then most likely they are coming to kill you.
Know your role: If you're the tank, be the tank. Protect your squishies, stay close to them and protect them. They are usually your carries and thats how you will win the teamfights. Rally your team, don't be afraid to boss them around. "All push mid" "Group up". Late game power is all yours as the tank, you engage, you chose where and when to engage and where to push. Without you your team is pretty useless. If you're the carry, stick to the tank farm within moderation and keep an eye on the map. You're who they're disappearing to come kill. Stick to the brush when teamfights engage so they can't come in and just ice you and run away. Don't reveal yourself till some spells have been cast and they have already begun focusing another less squishy player. If you're support, keep the tank alive and watch your squishys.
Don't wander: If its late game and you're all level 18 its time to be a team, move together, stay close. Don't over extend and wandering also involves watching the maps. If the enemy team is pushed into their base and fending off super mionons go steal their camps, but just always farm in moderation and never over extend.
Know your enemy: Once again coming back to knowing the champions. If you're the tank and the majority of the enemy team damage dealers are ability power then stack magic resist, if you're a carry and a majority of the enemies are tanky high health grab a madreds and tear them down. Don't trust mobafire 100%. No build is gaurenteed every time. I say this all the time, LoL is dynamic and should be played as such. Build your build around the enemy team and what your team needs from you.
Also a side-note worth mentioning, once you can buy tier 3 runes build some pages for your favorite champions. Try to make ambiguous pages that can apply to for example all AD carries or all tanks. Some good purchase starting points are armor seals, armor pen marks, and magic pen marks. You can use them for basically any character.
I think the best example of someone that falls off late game is AP Sion. Early game he is an absolute terror and can destroy just about anyone; late game he struggles with relevance.
@Mario's answer suggests that this is due to poor scaling, so lets take a look at Sion's abilities and base stats.
Cryptic gaze: 70 / 125 / 180 / 240 / 300 (+ 90% AP)
Death's Caress(dmg and shield strength): 100 / 150 / 200 / 250 / 300 (+ 90% AP)
A .9 ratio is nothing to scoff at. For comparison, Karthus' Lay Waste (single target) and Requiem have a .6 ratio and his defile is only at .25.
Let's compare sion's base stats to mordekaiser's.
SION
- HEALTH 403 (+104)
- ARMOR 17.75 (+3.25)
- MAGIC RES. 30 (+1.25)
MORDEKAISER
- HEALTH 421 (+80)
- ARMOR 15 (+3.5)
- MAGIC RES 30 (+1.25)
As you can see their level 1 base stats are extremely similar, assuming they're fighting an AP base stats would tend to suggest that Mordekaiser would be stronger early. At 18, Sion has 390 additional base health over Mordekaiser - suggesting that Sion would be stronger than Morde late game. This, however, is clearly not the case.
So then what exactly makes Sion fall off late game? His kit. Sion is extremely good early because he can effectively stun and shield burst to kill most champions. Once late game hits champions will generally (either naturally or build to be) tankier and be able to withstand this combo and peeling becomes much more effective. (ie, the ADC can lifesteal a respectable amount during CC)
Ultimately, the key to any champion falling off towards late game is their inability to no longer burst a target down and their impact in team fights is greatly reduced because of it. This seems to be the case to me with any early game champ (shaco, lee sin, renekton, etc).
(stats taken from league of legends wiki)
Best Answer
From my experience playing against and as him, he falls of due to his build, his team fighting abilities, and his abilities failure to scale well into late game. This doesn't mean that a Lee Sin can't own late game some games, just if the teams have comparable setups and the games pretty even.
Usually he's built pretty tanky with some decent damage, but no crazy damage, crit, or attack speed. This makes him a good initiator and good at chasing down people, but he doesn't have the cc or survivebility to tank or instigate team fights, and doesn't do enough damage to burn through enemy champs quickly.
I wouldn't even say he's bad late game, it's just that his play style has to change. You go from instigating fights and fearing no one, to waiting till a team fight starts and them jumping on the carrys in the back.
Lee Sin is a shit load of fun though, just loses that unstoppable feeling after you get 30-40 mins in...