A quick Google tells me "Because water boils at a lower temperature at high altitudes". But I am not being able to understand this answer and fill-in the gap. Like, how does an egg cook in the first place? What does it mean when we say that water boils at lower temperature? In fact, I would have thought otherwise that since water is able to boil at low temperature itself (which will be reached sooner), the egg will be able to cook sooner.

What is wrong with this reasoning?