This is my pet peeve topic because this alleged need for "trust" has become so ubiquitous. It's like a meme that is parroted blindly. At a closer look, it's in my opinion not so clear whether trust should play any substantial role in dealing with potential knowledge and insight, especially since the term "expert" is interpreted ranging from "a guy who wrote an M.A. thesis about some related topic" to "internationally renowned researcher with 100+ publications in a specific, narrow subtopic of a field who is highly estimated by their peers and has served in various panels to inform decision makers."

Generally, science is not built on trusting others, it's built on being skeptical.

What people miss is not trust in alleged or real experts, what they lack is intellectual humility and the ability to determine when someone else likely knows more about a topic than you. That doesn't mean that they have to trust that person's opinion about the matter, it just means that they should have a tendency to believe that person's informed opinions more easily than their own uninformed preconceptions.

For example, people shouldn't trust authorities and the experts these rely on. It's perfectly fine if they constantly scrutinize these authorities. However, they should be able to recognize when they clearly know less then these and act accordingly. Whether that means remaining agnostic or acting upon the expert's knowledge (even if it is taken with a pinch of salt) depends on the decision situation, how much need for action there is.

By the same token, you should also not trust text books. You should be aware of your own limitations while trying to check the information in the book. When you know you don't have enough knowledge to check it, you're going to have to trust some of the content. The more you build up knowledge, the more you can check the book, and at some point you will find errors in it.