If you watch enough YouTube videos of repairing old micros — and even ones that aren't that old (486s didn't come out that long ago, did they?) — you'll see a lot that have suffered some sort of damage due to storage over the years. Some damage, such as corrosion due to battery leakage or cracked circuit boards, is not due to climate. However some, such as oxidized contacts or corroded circuit board traces, is.
Old mainframes and minicomputers have a lot of points of failure compared to micros. Their construction often involves a lot of socketed components, plugs and connectors, and wire-wrapped connections. Each of these is an unprotected point where corrosion can occur. Repeated heating and cooling cycles can unseat socketed components and connectors, or weaken iffy solder joints. Humidity can corrode exposed wire, pins, and traces. Many old printed circuit boards do not have protective solder mask layers, which leaves them exposed as well. And there are a lot of boards, discrete components, and wired connections, which means a lot more pieces that can go bad.
Contrast this to the construction of most micros: more modern circuit boards, often (but not always, depending on age) with a protective solder mask layer. Many components are directly soldered to the board, and there are often only one or two boards. Almost all the connections are internal to the circuit boards. There are fewer discrete components, because there are more complex integrated circuits. There are far fewer exposed connections that can flake out, or components to be damaged.
Also, don't discount age as a factor. Depending on the computer, the quality of construction of some of the components may simply not have been as good because no one yet knew how to make them better, so ambient conditions may damage them too.
Saying "will destroy" may be something of an exaggeration depending on climate. A moderate or even hot, fairly dry climate will likely be reasonably kind to electronics if they're sheltered. A cool and damp one without too much temperature variation may not be a death sentence. In my opinion, a warning like that is still a good idea to someone who doesn't know the ins and outs of old electronics, since it will prevent disappointment and wasted money.