nontrivial theorems with trivial proofs

Bertrand Russell proved that the general set-formation principle known as the Comprehension Principle, which asserts that for any property $\varphi$ one may form the set $\lbrace\ x \mid \varphi(x)\ \rbrace$ of all objects having that property, is simply inconsistent.

This theorem, also known as the Russell Paradox, was certainly not obvious at the time, as Frege was famously completing his major treatise on the foundation of mathematics, based principally on what we now call naive set theory, using the Comprehension Principle. It is Russell's theorem that showed that this naive set theory is contradictory.

Nevertheless, the proof of Russell's theorem is trivial: Let $R$ be the set of all sets $x$ such that $x\notin x$. Thus, $R\in R$ if and only if $R\notin R$, a contradiction.

So the proof is trivial, but the theorem was shocking and led to a variety of developments in the foundations of mathematics, from which ultimately the modern ZFC conceptions arose. Frege abandoned his work in this area.

The additivity of expected value is absolutely trivial to prove, but (I think) mind-blowing that it is true.

Also, the fact that (finite) sums/products of vector spaces are isomorphic. Extremely easy, but amazingly powerful. It is the reason we can do linear algebra with matrices.

A nontrivial geometric theorem of the type you are looking for may be the Desargues theorem:

If two triangles are in perspective then the intersections of their corresponding sides lie on a line.

In three dimensions there is a trivial visual proof:

_{(source: schillerinstitute.org)}

But the theorem is nontrivial because there is no projective proof in two dimensions -- there are projective planes in which the theorem does not hold.

The plot thickens when one investigates the algebraic reasons for this. Hilbert discovered that the Desargues theorem is equivalent to associativity of the underlying coordinate system. So, a projective plane with octonion coordinates, for example, does not satisfy the Desargues theorem.

Addendum. In answer to your first question, the quote is a garbled version of Grothendieck, quoting Ronnie Brown quoting J.H.C. Whitehead. I found it on p.188 of the PDF version of Récoltes et Semailles. Translating back into English, it becomes:

... the snobbery of the young, who think that a theorem is trivial because its proof is trivial.