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Proven, but nobody knows why
The continuum hypothesis asks whether anything sits between the size of infinity of the natural numbers and the size of infinity of the real numbers. The answer is both yes and no, depending on your axioms, which puts it outside the usual axiom systems. The method that shows this is called forcing, and it is really hard to understand.
Timothy Chow opened an expository paper on forcing by pointing out that everyone knows what an unsolved research problem is. He proposed the idea of an unsolved expository problem: something we have proven but do not really know why it is true. He then offered a partial solution to it.
Recent mathematics has plenty of cases where the reach exceeded the grasp and results were proven long before they were understood.
How did Cohen show the hypothesis unprovable?
In 1963, Paul Cohen built a new technique, forcing, that starts from a model of set theory and adds new sets to it in a controlled way. He used it to construct a model of the standard axioms in which the continuum hypothesis fails. Kurt Godel had shown in 1940 that a model exists in which it holds. Together the two results show the axioms alone settle neither answer. Cohen received the Fields Medal in 1966 for this work.
Which other proofs came before understanding?
The four color theorem was proven in 1976 by Kenneth Appel and Wolfgang Haken, who used a computer to check nearly 2,000 configurations. Many mathematicians accepted the result and still wanted a proof that explains why four colors always suffice. A later verification by Georges Gonthier in 2005 confirmed it inside a proof assistant, but a short human-readable argument remains absent.
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