My question is, how do they manage to access content that's behind someone's paywall, as here?
Paywalls are complicated and there is no single answer. In particular I can’t tell anything about that specific pair (Wayback Machine + Wired).
For start, not all clients are served the same. It’s often dynamically determined. Legal limitations, providing samples to lure the target, geodiscrimination, whatever information they already have about you, blocking the unwelcome, political affiliations, agreements with 3rd parties, pretty much anything can affect if you see a paywall or not. For example I never seen a paywall on Wired. I can read the linked article with no issues either. So it’s possible that solely determined a crawler was able to access content.
Many paywalled sites intentionally provide full content to search engine crawlers to lure audience or limit visibility of the competition. While Wayback Machine isn’t a search engine, it ends up in many classifiers marked as such. Most companies slurp those indiscriminately or delegate the task to other parties. It’s not unexpected for a crawler to be served as a search engine even while not being one.
All those publishers must know about this. What's the deal here?
Must they? Don’t overestimate general population’s knowledge and abilities.

Ignoring that and assuming they know: the knowledge does nothing by itself. The action is needed. But what would be the action? Archive doesn’t own assets big enough to make profit from suying them. Not big enough for
a business with a $2bn/yr revenue, and even smaller when expenses and risks are taken into account. DMCA-ing entries? That requires hiring people Which would only be viable, if losses are considerable. And what losses they may have from the 0.01% outliers going to Wayback Machine, outliers who are unlikely to ever subscribe anyway? Blocking Wayback Machine? The same problem. And imagine all the meetings needed, all the decisions on each management level, consultations regarding brand image impact, determining what level of financial harm will that bring regarding lower exposure on Wikipedia, hiring experts to determine security and reliability impact of changing access policies, having somebody to pay to maintain that change? C’mon, be reasonable.
Finally: companies are not conscious beings. They have legal personhood, but they aren’t literal persons. They don’t have personality, they don’t have thoughts, beliefs, fears or desires. It may be a convenient metal shortcut to say “they know” or “they may do this or that.” But in the end, who are the they? The they, who have actual thoughts, beliefs, fears and desires? It’s not the organisation. It’s the people hired. Do you think some Conde Nast manager, facing a paywall, is subscribing? Or, if only they can use computers well enough, goes just like you to Wayback Machine? Unless forced to by the work obligations, it’s not in their interest to block anything.