This line that you are citing comes from "The Open Window," by Saki. The narrator says this with reference to Framton Nuttel. The narrator uses these lines to describe Nuttel when Nuttel starts to tell Mrs. Sappleton about his medical condition. He tells her what the doctors have told him he should and should not do.
As for meaning, to labor under a delusion is to have the wrong idea about something. So Nuttel believes something that is not correct. This delusion is "tolerably widespread." This means that quite a few people think that way. So Nuttel's belief is one that many people have.
Once you understand that, the rest is relatively easy -- he believes that people who do not know him well (like Mrs. Sappleton) would be eager to hear ("hungry") all about ("the least detail") his medical condition.