More often than not, it's by getting an early lead. Getting an early lead on one character who scales well often means that you have a chance no matter what, and some people will recognize that. I remember a game where I was playing Twitch bot lane with a Nami support. The Nami initially hovered Twitch support, but I wanted to play my main and this person gave him to me. However, when we got in-game, this Nami started legitimately inting. She did not want to be in that game. However, she took long enough for the enemy bot lane to kill that I was able to get fully stacked passive and a double kill on their bot lane. Next time she came to lane, it happened again, and again. She finally stopped inting and starting playing seriously when I was 8/0. At that point, I was so fed that even this Nami, who seriously didn't want to be in that game to the point where she was inting at level 1, started playing legitimately seriously and trying to win. Why? She saw an opportunity to win. If anyone sees any chance that you can carry, they will play seriously. Most of the time, however, lower tier players can't see a way to win if nobody on their team has a lead, even if the enemy team doesn't scale anywhere near to as well as theirs.
I had another game that I want to use as an example of something probably a bit more relevant. I was playing Kha'Zix mid lane. I got a huge early lead against my lane opponent, but despite my lead, the rest of my team was feeding to the point where we were 15,000 gold down at 20 minutes. There was a point where we had a surrender vote be 3/1 and we were able to convince the 5th player that the enemy team was starting to throw after a couple of teamfights where the enemy team played poorly and let me get to their carries. The game went to 75 minutes, but I was able to hard carry that game. After we won teamfights played poorly by the enemy team, I would make a bunch of calls that would give us different objectives (typically a turret + a dragon or 2 turrets). If you can get strong enough to win teamfights for your team and get said teamfights to happen, you can carry your team in any game so long as you take an objective afterward.