Pepijn explained this well, but I'd like to add an example of what bounded accuracy is meant to avoid.
In fourth edition, to-hit bonuses, armor class, proficiency bonuses, and so on scaled roughly linearly with character level. This is sometimes called "level scaling" and meant that a first-level fighter had about the same chance of hitting a first-level monster as a twentieth-level fighter hitting a twentieth-level monster.
It also meant that a twentieth-level fighter would have a 100% chance of hitting a first-level monster, and a first-level fighter would have a 0% chance of hitting a twentieth-level monster*.
* Technically not quite 100% and 0% because of the crit rules; if you roll a nat 20 and it's still not enough to beat the target's AC, for example, you still score a normal hit. But that's not important here.
So if your 20th-level party wants to go take out a bunch of kobolds and goblins, you can't use the stat blocks you used at lower levels—there would be no risk, it wouldn't be especially exciting or fun. Instead, there are alternate stat blocks for use at higher levels, which have appropriately scaled bonuses and defenses, but are marked as "minions" (they have only one hit point but never take damage from missed attacks).
This worked well enough. But it could also be unsatisfying, because while the raw numbers went up steadily as you gained levels, the actual mechanics of combat didn't really. You still had to roll about the same on the d20 to hit, and you'd take out about the same percentage of the enemy's hit points. And, more importantly, you had to work to avoid falling behind the curve for any reason—you needed increasingly powerful magic items just to maintain the same odds of hitting a level-appropriate monster. If the wizard picked up a crossbow at first level but never put any effort into getting better with it, they would get worse at using it against basic kobolds, while their spells still have about the same odds of hitting.
5e decided to avoid this. If you're in a one-on-one fight against a single monster of appropriate CR, your odds of hitting will stay about the same. But now, even a 20th-level character should have a chance of missing a 1st-level goblin from the start of the game, or getting hit by them, and a higher-level character shouldn't get any worse at things they don't focus on upgrading. To accomplish this, all bonuses and targets are kept within a certain range regardless of level.