What do the numbers G_4 and G_6 of a lattice actually measure?

As far as I know $G_4$ and $G_6$ don't have a direct geometric interpretation of the type you are looking for. Rather, they appear as coefficients in the algebraic equation for $\mathbb C/\Lambda.$ More precisely, the pair $(\mathbb C/\Lambda, dz)$ consisting of the complex torus $\mathbb C/\Lambda$ and the everywhere-holomorphic differential form $dz$ is isomorphic to the pair $(E,dx/y),$ where $E$ is the smooth complex projective curve cut out by the (homogeneous equation associated to) the equation $y^2 = 4 x^3 - 60 G_4(L) x - 140 G_6(L).$ (Here the letter $E$ is for "elliptic".)

As you probably know, this is more or less the content of the theory of Weierstrass's elliptic functions.

In summary: lattices in $\mathbb C$ are the same thing as elliptic curves over ${\mathbb C}$ equipped with a choice of non-zero holomorphic differential (via $\Lambda \mapsto (\mathbb C/\Lambda, dz)$, and the quantities $G_4$ and $G_6$ give an explicit formula for this correspondence, by describing the coefficients of the algebraic equation for the corresponding elliptic curve.