Suppose we fit a simple mixed / multilevel model with no predictors, which some call a variance components model:
$$ y_{ij} = \beta_0 + u_j + e_{ij} $$
where $u_j$ are the level-2 residuals and $e_{ij}$ are the level-1 residuals, and we obtain estimates, $\hat{\sigma_u^2}$ and $\hat{\sigma_e^2}$ for the variance of $u_j$ and $e_{ij}$ respectively. I get it that
$$ \nu = \frac{\hat{\sigma_u^2}}{\hat{\sigma_u^2} +\hat{\sigma_e^2}} $$ is the proportion of total variance in the response due to between-level-2 units, and so it makes sense that this is called the variance partition coefficient.
What I don't get is why $\nu$ can also be interpreted as the correlation between response scores for 2 randomly selected observations from within the same level-2 unit (ie the intra-class correlation). It makes sense that $\nu$ should be related to this correlation, but why should be be exactly equal ?