ICC as expected correlation between two randomly drawn units that are in the same group In multilevel modelling the intraclass correlation often gets calculated from a random-effects ANOVA
$$ y_{ij} = \gamma_{00} + u_j + e_{ij} $$
where $u_j$ are the level-2 residuals and $e_{ij}$ are the level-1 residuals. Then we obtain estimates, $\hat{\sigma}_u^2$ and $\hat{\sigma}_e^2$ for the variance of $u_j$ and $e_{ij}$ respectively, and plug them into the following equation: 
$$
ρ = \frac{\hat{\sigma}_u^2}{\hat{\sigma}_u^2 +\hat{\sigma}_e^2} 
$$
Hox (2002) writes on p15 that 

The intraclass correlation ρ can also be interpreted as the expected
  correlation between two randomly drawn units that are in the same
  group

There's a question here that asks an advanced question (why it is exactly equal to this instead of approximately equal) and gets an advanced answer.
However, I wish to ask a much simpler question. 
Question: What does it even mean to talk about a correlation between two randomly drawn units that are in the same group? 
I have a basic understanding of the fact that the intraclass correlation works on groups and not on paired data. However, I still don't understand how the correlation could be calculated if all we had was two randomly drawn units from the same group. If I look at the dot plots on the Wikipedia page for ICC, for example, we have multiple groups and multiple points within each group.
 A: It may be easiest to see the equivalence if you consider a case where there are only two individuals per group. So, let's go through a specific example (I'll use R for this):
dat <- read.table(header=TRUE, text = "
group person   y
1     1        5
1     2        6
2     1        3
2     2        2
3     1        7
3     2        9
4     1        2
4     2        2
5     1        3
5     2        5
6     1        6
6     2        9
7     1        4
7     2        2
8     1        8
8     2        7")

So, we have 8 groups with 2 individuals each. Now let's fit the random-effects ANOVA model:
library(nlme)
res <- lme(y ~ 1, random = ~ 1 | group, data=dat, method="ML")

And finally, let's compute the ICC:
getVarCov(res)[1] / (getVarCov(res)[1] + res$sigma^2)

This yields: 0.7500003 (it's 0.75 to be exact, but there is some slight numerical impression in the estimation procedure here).
Now let's reshape the data from the long format into the wide format:
dat <- as.matrix(reshape(dat, direction="wide", v.names="y", idvar="group", timevar="person"))

It looks like this now:
   group y.1 y.2
1      1   5   6
3      2   3   2
5      3   7   9
7      4   2   2
9      5   3   5
11     6   6   9
13     7   4   2
15     8   8   7

And now compute the correlation between y.1 and y.2:
cor(dat[,2], dat[,3])

This yields: 0.8161138
Wait, what? What's going on here? Shouldn't it be 0.75? Not quite! What I have computed above is not the ICC (intraclass correlation coefficient), but the regular Pearson product-moment correlation coefficient, which is an interclass correlation coefficient. Note that in the long-format data, it is entirely arbitrary who is person 1 and who is person 2 -- the pairs are unordered. You could reshuffle the data within groups and you would get the same results. But in the wide-format data, it is not arbitrary who is listed under y.1 and who is listed under y.2. If you were to switch around some of the individuals, you would get a different correlation (except if you were to switch around all of them -- then this is equivalent to cor(dat[,3], dat[,2]) which of course still gives you 0.8161138).
What Fisher pointed out is a little trick to get the ICC with the wide-format data. Have every pair be included twice, in both orders, and then compute the correlation:
dat <- rbind(dat, dat[,c(1,3,2)])
cor(dat[,2], dat[,3])

This yields: 0.75.
So, as you can see, the ICC is really a correlation coefficient -- for the "unpaired" data of two individuals from the same group.
If there were more than two individuals per group, you can still think of the ICC in that way, except that there would be more ways of creating pairs of individuals within groups. The ICC is then the correlation between all possible pairings (again in an unordered way).
