In other words, sound (seismic waves) travels much faster in one direction than the other through the inner core. That's true of most rocks to some degree, and it implies that the crystalline iron in the inner core is mostly aligned in a similar direction. But that is at the core (pun intended) of all of this.

So the "fast direction" has subtly changed over time based on the data we have. That's the "the Earth's inner core rotates differently than the rest" part. But we're mostly basing that on travel times in each direction (it's more complex than that - more in a bit). The differences "more fast stuff" and "less slower stuff" are hard to distinguish precisely, though they can be distinguished because of effects that occur at the boundary between different velocity + density bodies. It's also harder because the outer core is liquid and removes a key source of information coming from wave interactions at those boundaries (shear waves).

This is basically doing a lot of clever reprocessing of old data to carefully look data after corrections for the moderately-well-constrained rotation of the inner core. Rotation of the inner core can't explain all of the differences, so another thing that might cause it is changes in the shape of the boundary between the inner and outer core. It's also possible it's noise, though presumably the authors investigated that part carefully (haven't read the scientific article, but the primary author is a very well known person in the field, so the analysis is likely very sound).

There are always alternate explanations, though. Changes in shape on the order called for here do need an explanation via geological processes. Kilometer scale changes in a decade are difficult to immediately explain, though not impossible. I have no doubt the analysis is sound, but from a geologic perspective, this (and previous) work raises a lot of interesting questions.

https://xkcd.com/1387/