Picking up on adrianN's comment[0], when you have a collection of nodes and start connecting them at random, initially they are all disconnected (obviously) and any two that you pick are likely not to have any edges. This in the early stages, your graph is isolated nodes and isolated edges.
After a while, by chance, you happen to join an existing edge to a node. That component now has three vertices, and is 50% more likely to be chosen at random than the isolated edges.
There comes a point where you join two non-trivial components, and before long you reach a tipping point. Suddenly nearly every node you choose already belongs to a component, and that component starts vacuuming up everything.
Thus we have the emergence of "The Giant Component". This transition is sharp and well-studied. Whether you think of it as "obvious" depends on how much you study these things. I seem to recall that there is a major result that says that all first-order predicates have these threshold emergence properties, but it's been too long (30 years) since I studied this, and I could be wrong. I may be able to find some references if people really want me to.
[0] https://news.ycombinator.com/item?id=6546978