It means that the difference between strengths often decreases between X champion and the enemy team. Usually this is because the flow of the game transfers from skirmishes/1v1's and laning -> teamfights/objective control as the game progresses, and they both require a different set of strengths to succeed in.
Take Pantheon as an example:
His kit is great at harass and providing a constant threat to a single target near him. His range limits aren't hindered in lane because his opponent is forced to come close to his ranges in order to grab cs, while his passive blocks an auto which is incredibly significant if there is only one source of damage to threaten him (aka his enemy laner), especially because attack speed is low in the early game as well. This is were it's maximum damage prevention can be found.
Come to late game where teamfights and objective control is prioritized more, you'll see that Pantheon's kit tends to "fall off". This is because there are multiple sources of damage that will be directed at a diver like Pantheon, so the significance of his passive is reduced dsignificantly (and higher attack speeds in the late game mean that the dps that it blocks is negligible as well). His single-target damage is preventable due to there now being a frontline to tank his damage and zone him out from the squishies he would like to dive, which is not a factor in the early game because nobody has built resistances yet and Panth starts out with high base stats. There are also champions that can peel him off as well. The lack of a needed resource (cs) makes his range limitations much more significant because now he is the one being punished for getting too close, unlike early game where his laner is the one to be punished.
"Falling Off" is usually attributed to a champion's kit's limitations becoming more apparent or highlighted in the late game where new rules or aspects of the game is introduced.
This is the reverse of a champion who is known for a strong late game. I won't be as detailed as before, but take Vayne as an example:
Her W is a huge part of her kit's damage. Early game, when her attack speed is low, her damage is inherently lower because she lacks the items to build attack speed. Late game when she has max attack speed, she now makes full use of her W and so her damage output is at it's maximum.
So unlike Pantheon, who loses efficiency with his kit, Vayne (and most adc's who are designed for consistent damage) scale well into late game because they gain efficiency in their kit with attack speed.