What's the "geometry" in "geometric multiplicity"?

@Bruno is essentially correct. It's important to see that geometric multiplicity is meant to be distinguished from algebraic multiplicity of eigenvalues, the latter being the total number of times an eigenvalue occurs as a root of the characteristic equation. An analogy can be made with roots of any polynomial. For example, $x^2 + 2x + 1$ has a single root $-1$ of multiplicity 2. Algebraically, there are always two roots for a quadratic (at least over $\mathbb{C}$), and in this case, those roots are $-1$ and $-1$. But (geometrically) there is only one $x$-intercept for the function $y = x^2 + 2x + 1$.