Rubberband AI, more favorably called "dynamic game difficulty balancing", essentially tries to emulate an age old parenting concept and applies it to gaming: No matter how good or how bad you are at something, it will always try to encourage you. Whether that works is another question.
The principle, as far as I know, is derived from racing games, and is also best explained in this context: Imagine you had a rubberband strapped around you that was connected to your opponents. On the one hand, no matter how fast you go, your opponents will never be very far behind - in fact, the more distance you have compared to them, the faster they go. On the other hand, even if you have an accident and lose a lot of time, your opponents won't be able to completely escape you, and you will get the chance to literally "bounce back" quickly.
Rubberband AI is of course applicable to almost any other type of game with some kind of NPC: In fighting/shooting games, it might manipulate the strength and number of your opponents based on your skill, in point-based games it will simply make the AI reach scores very similar to yours. Other aspects of the game are also easily scaleable according to player skill, such as player strength, the amount of available bonuses/upgrades, game speed, or the amount of time available.
One game that took rubberband AI to the extreme was Need for Speed: Underground. I was once so fed up of the opponents catching up with ease that I stepped on the brake and simply continued to drive around at a measly 5-15 mph for a while, then sped up just a little bit. Around the next corner, the opponents were literally just waiting for me. Thus, you could in theory take three minutes for a race and lose, but take half an hour for the same race, and win it.
Therefore, in essence, rubberband AI is a lazy solution to the problem of writing an actually good AI and a level design that progressively challenges the player on its own terms. While a rubberband AI can ease the process of getting familiar with a game and its mechanics, the real problem is that the game loses its long-term payoff: For long-term investment in a game, it is important that improvements in player skill correspond with progress in the game. With a rubberband AI, player skill and its improvement will be largely irrelevant to progress in the game.
Iron Noder Challenge 2017