TL;DR. Saying that enchanted armor breaks bounded accuracy is somewhat incorrect, but excessively high AC does break bounded accuracy and limiting the availability of enchanted armor is the most convenient solution.
Excessively high AC breaks bounded accuracy
To understand why, you need to understand that bounded accuracy exists to achieve a list of desirable goals. Breaking bounded accuracy doesn't necessarily mean that the game becomes unplayable, it just means one or more of those desirable goals has not been met. However, a lot of DM resources are written with the assumption that bounded accuracy is respected, hence when it is broken some of those resources become unreliable or incorrect.
The article "Bounded Accuracy" on Legends and Lore (an official D&D website) by Rodney Thompson (who, at the time, was a WOTC employee and is credited with Rules Development on the 5e PHB, among credit for other WOTC products) does a great job of explaining what those desirable goals are.
A few of those goals are relevant to this question. Among them, the most relevant is the following:
The DM's monster roster expands, never contracts. Although low-level characters probably don't stack up well against higher-level monsters, thanks to the high hit points and high damage numbers of those monsters, as the characters gain levels, the lower-level monsters continue to be useful to the DM, just in greater numbers. While we might fight only four goblins at a time at 1st level, we might take on twelve of them at 5th level without breaking a sweat. Since the monsters don't lose the ability to hit the player characters—instead they take out a smaller percentage chunk of the characters' hit points—the DM can continue to increase the number of monsters instead of needing to design or find whole new monsters. Thus, the repertoire of monsters available for DMs to use in an adventure only increases over time, as new monsters become acceptable challenges and old monsters simply need to have their quantity increased.
In other words, the monster's usefulness should depend mostly on their damage, not on their bonus to hit, so that the DM's roster (the list of useful monsters) never shrinks even as the party becomes more powerful.
A monsters that only has a 5% chance to hit, is not useful. The DM would have to run an impractically large number of said monsters for them to have a significant impact on a combat. Even monsters with a 15% chance to hit are borderline impractical: there's a very real chance such a monster will never hit over the course of several turns and that its only impact on the combat is to waste time.
Imagine that half of the party has an excessively high AC, for example 26 AC. In this situation, the DM's roster shrinks to exclude monsters with +6 bonus to hit or lower and even monsters with a +8 bonus to hit are borderline impractical. In other words, the desirable goal has not been met and bounded accuracy has been broken. It just so happens that +3 full plate plus +3 shield is 26 AC.
Excessively high AC is desirable from a player's perspective
There are a lot of ways to increase AC and most of them stack:
- +X armor, or alternative items such as Bracers of Defense.
- +X shields.
- spells such as shield, shield of faith, and haste.
- features and feats such as Cutting Words, and Defensive Duelist
- half or three quarters cover
- and more.
If a majority of the those things are easily available, then it's possible for PCs to have excessively high AC on a regular basis.
And it's not just possible, it is also desirable from the player's perspective, because naturally the player doesn't want their character to be hit. Moreover, increasing AC even yields increasing returns, for example against a +9 to hit: increasing AC from 15 to 16, decreases damage by 1/15; increasing AC from 25 to 26, decreases damage by 1/5.
This is a problem, because it means the game provides both the means and the incentives to break bounded accuracy in the context of AC.
Limiting magic items is the most convenient solution
Of all the things which increase AC, magic items are the only ones entirely under the DM's control, hence the easiest way to preserve the AC's bounded accuracy is to limit the availability of those magic items.