Minkowksi's Convex Body Theorem is evidently pretty powerful, as it yields swift proofs of Fermat's Two Square and Lagrange's Four Square Theorems. Also, Minkowski's bound on class number and the proof of the Dirichlet Unit Theorem.
Is there some intuition as to why looking at Minkowski's Theorem/lattice geometry should tells us so much about the integers?
If we have an algebraic number field $K$ we can consider the real places and the complex ones i.e. embeddings $\sigma: K\to \mathbb{C}$, if $\sigma(K)\subseteq \mathbb{R}$, we say $\sigma$ is a real-place. Each of these places defines a Archimedean absolute value on $K$. Now form the product $\prod_{\sigma} K_{|\cdot |_{\sigma}}$ where $|\cdot |_{\sigma}$ is the induced absolute value and $K_{|\cdot |_{\sigma}}$ is the completion. The map $K\to \prod_{\sigma} K_{|\cdot |_{\sigma}}$ sends $K$ to a lattice, each completion is isometric to $\mathbb{R}$. Perhaps we can then say that if we known something about the lattice, we know something about the Archimedean absolute values, which in turn gives us information about $K$?