Why should we prove obvious things? Obviously, there are obvious things in mathematics. Why we should prove them?


Prove that $\lim\limits_{n\to\infty}\dfrac{1}{n}=0$?
Prove that $f(x)=x$ is continuous on $\mathbb{R}$?
$\dotsc$


Just to list few examples.
Because sometimes, things that should be "obvious" turn out to be completely false.  Here are some examples:


Switching doors in the Monty Hall problem "obviously" should not affect the outcome.
Since Gabriel's horn has finite volume, then it "obviously" has finite surface area.
"Obviously" we cannot decompose sphere into a finite number of disjoint subsets and reconstruct them into two copies of the original sphere.
Since the Weierstrass function is everywhere continuous, then "obviously" it must have at least a few differentiable points.


Of course, mathematics has shown that switching doors is to the player's advantage, that Gabriel's horn actually has infinite surface area, that you can indeed get two copies of the original sphere (see Banach-Tarski paradox), and that the Weierstrass function is everywhere continuous but nowhere differentiable.  The point being, there are many things out there which are "obvious" but actually turn out to be entirely counterintuitive and opposite what we would otherwise expect.  This is the point of rigor: to double check and make sure our intuition is indeed correct, because it isn't always.
In my opinion, only axioms should be treated as obvious, above all while a theory is being explained to others. I think it is simply immoral for a mathematician who is writing a proof of a proposition in a book not to give every smallest detail in the chain of logical inferences, skipping the task of making the subject perfectly clear through a lazy abuse of the magic words trivial, obvious, exercise. If one wants to write mathematics for others, he must be as clear and rigorous as possible, at least while developing theory and examples. If instead he's too lazy to bear such a sacred task, then he should not even begin to write mathematics, because the essence of mathematics is in its logical clarity, and not in a pseudomystic vagueness - alas, many important books of great mathematicians are "written" in this terrible style! -.