A simple classical example where this is true is a repetition code. If you represent 1 as 11...1 and 0 as 00...0, randomly flip some bits, and then recover the logical state with majority voting, the probability of a logical error occurring shrinks exponentially as you add more bits to it. This is because a logical error requires flipping at least half the bits, and the probability of that happening decreases exponentially.
The error correcting code used in this work also has a nice intuitive explanation. Imagine a 2D grid of bits, visualized as pixels that can be black or white. Now imagine drawing a bunch of lines of black pixels, and enforcing that lines can only end on the top or bottom boundary (loops are allowed). If there is an even number of lines connecting the top to the bottom, we call that logical 0, and if there is an odd number of lines, we call that logical 1. This again has the property that as you add more bits, the probability of changing between logical 1 and 0 gets exponentially smaller, because the lines connecting top to bottom get longer (just like in the repetiton code).
This code also has the nice property that if you measure the value of a small patch of (qu)bits, there's no way to tell what the logical state is. This is important for quantum error correction, because measurement destroys quantum information. So the fact that local measurements don't reveal the logical state means that the logical state is protected. This isn't true for the repetition code, where measuring a single bit tells you the logical state.
The error correcting code used in this work also has a nice intuitive explanation. Imagine a 2D grid of bits, visualized as pixels that can be black or white. Now imagine drawing a bunch of lines of black pixels, and enforcing that lines can only end on the top or bottom boundary (loops are allowed). If there is an even number of lines connecting the top to the bottom, we call that logical 0, and if there is an odd number of lines, we call that logical 1. This again has the property that as you add more bits, the probability of changing between logical 1 and 0 gets exponentially smaller, because the lines connecting top to bottom get longer (just like in the repetiton code).
This code also has the nice property that if you measure the value of a small patch of (qu)bits, there's no way to tell what the logical state is. This is important for quantum error correction, because measurement destroys quantum information. So the fact that local measurements don't reveal the logical state means that the logical state is protected. This isn't true for the repetition code, where measuring a single bit tells you the logical state.