The article uses "planning" to mean "classical planning", which is a very specific thing, although it's such a fundamental concept in AI research that it is very difficult to find a simple definition (there's a lot of useless stuff on the internet about it, like tutorials that don't explain what it is they're tutorial-ing, or slides that don't give much context).
Even the Wikipedia article is not very well written. I followed this link to one of its references though and there's an entire textbook, available as a free pdf:
https://projects.laas.fr/planning/
In general, classical planning is one of those domains where GOFAI approaches continue to dominate over nouveau AI, statistical machine learning-based approaches. You'll have to take my word for that, though, because that's what I know from experience, and I don't have any references to back that up. On the other hand, if it wasn't the case, you wouldn't see papers like the one linked above, I suppose.
To clarify, the paper above makes it clear that LLMs, for one, are useless for planning but at least they can translate between natural language and PDDL, so that a planning problem can be handed off to a classical planning engine, that can actually do the job. How useful is that, I don't know. A human expert would probably do a better job of writing PDDL from scratch, but that's never explored in the linked article.