Geometric interpretation of Hölder's inequality

(Before getting to intuition: you've mention the proof by Young's inequality, but there is another nice proof that is a direct application of Jensen's inequality. Assume $\|f\|_p=\|g\|_q=1$ and $f,g>0.$ Then $(\int fg)^p=[\int g^q(fg^{1-q})]^p\leq \int g^q(fg^{1-q})^p=\int f^p=1.$ The inequality comes from the convexity of $x^p$ and probability measure $d\mu=g^qdx.$)

In any Banach space $V$ there is an inequality $|\langle x,f\rangle|\leq \|x\|_V \|f\|_{V^*}.$ This is almost a triviality, but it is a reflection of the geometrical fact that unit balls are convex. Vectors in the dual space $V^*$ represent hyperplanes that support the unit ball in $V.$

From the viewpoint of the geometry of Banach spaces, the intuition behind Hölder's inequality is that the dual space of $L^p$ has an integral representation as $L^q.$ If you take a point $f_0$ on the $L^p$ unit sphere, the supporting hyperplane is $\{f\mid\int fg=1\}$ where $g=\overline{f_0}|f_0|^{p-2},$ or simply $g=f_0^{p-1}$ with the assumption $f_0>0.$ For $\|f\|_p=1$ we get $\int fg\leq 1,$ which is Hölder's inequality.

Even if you had no idea what the dual space of $L^p$ might be, it would be natural to derive the convexity inequality $f^p\geq f_0^p+pf_0^{p-1}(f-f_0)$ - this is the equation for the supporting hyperplane at $x=f_0$ on the curve $x\mapsto x^p.$ Integrating gives $\int f^p\geq \int f_0^p+p\int f_0^{p-1}(f-f_0),$ which is a type of supporting hyperplane equation in $L^p.$ Rearranging and setting $f_0=g^{1/(p-1)}$ gives $\int f^p+(p-1)\int g^q\geq p\int fg,$ which is Hölder's inequality, after scaling to get $\|f\|_p=\|g\|_q=1.$ This argument is just Young's inequality in disguise but I hope the appearance of $q$ seems quite natural here.

For all these kinds of inequalities it is worth reading https://terrytao.wordpress.com/2007/09/05/amplification-arbitrage-and-the-tensor-power-trick/ if you haven't already.