“In seasons of pestilence some of us will have a secret attraction to the disease--a terrible passing inclination to die of it. And all of us have like wonders hidden in our breasts, only needing circumstances to evoke them.”
I think this quote means that even though something bad is going on around you, you feel a desire to want in on it. Even if getting in on this repudiated subject will get you into trouble, you have a need to be a part of it. I think that the second sentence is stating something similar to the first: it tells of a man's heart that will prove to want this consequential "thing", and that he only needs one incident to bring that want to the surface.
The connection I see in Heart of Darkness to this quote, is when Kurtz is obsessed with the ivory. He wants the ivory so bad that he will put his own health in danger by not taking care of himself. His obsession with the ivory causes his death, but it is what his heart wanted because it would gain him profit.
The first thing that came to mind when trying to relate it to another book we have read, is Lord of the Flies. I saw the "consequential thing" as being the power some of the boys wanted. None of the boys knew they wanted this power over their peers until they saw a chance to have it when the plane crashed. Even though terrible things came out of wanting this power, they couldn't stop themselves from wanting it because it was instilled in them.