The R model formula
lmer(measurement ~ 1 + (1 | subject) + (1 | site), mydata)
fits the model
$$ Y_{ijk} = \beta_0 + \eta_{i} + \theta_{j} + \varepsilon_{ijk} $$
where $Y_{ijk}$ is the $k$'th measurement from subject $i$ at site $j$, $\eta_{i}$ is the subject $i$ random effect, $\theta_{j}$ is the site $j$ random effect and $\varepsilon_{ijk}$ is the leftover error. These random effects have variances $\sigma^{2}_{\eta}, \sigma^{2}_{\theta}, \sigma^{2}_{\varepsilon}$ that are estimated by the model. (Note that if subject is nested within site, you would traditionally write $\theta_{ij}$ here instead of $\theta_{j}$).
To answer your first question regarding how to calculate the ICCs: under this model, the ICCs are the proportion of the total variation explained by the respective blocking factor. In particular, the correlation between two randomly selected observations on the same subject is:
$$ {\rm ICC}({\rm Subject}) = \frac{\sigma^{2}_{\eta}}{\sigma^{2}_{\eta}+ \sigma^{2}_{\theta}+\sigma^{2}_{\varepsilon}}$$
The correlation between two randomly selected observations from the same site is:
$$ {\rm ICC}({\rm Site}) = \frac{\sigma^{2}_{\theta}}{\sigma^{2}_{\eta}+ \sigma^{2}_{\theta}+\sigma^{2}_{\varepsilon}}$$
The correlation between two randomly selected observations on the same individual, and at the same site (the so-called interaction ICC) is:
$$ {\rm ICC}({\rm Subject/Site \ Interaction}) = \frac{\sigma^{2}_{\eta}+\sigma^{2}_{\theta}}{\sigma^{2}_{\eta}+ \sigma^{2}_{\theta}+\sigma^{2}_{\varepsilon}}$$
It seems you were confused by this being referred to as an "interaction" since it's the sum of individual terms. It's an "interaction" in the sense that it estimates the ${\rm ICC}$ corresponding to the blocking factor composed on the combination of Subject and site - it's important to note that you do not have to include some kind of "interaction" term between the factors to estimate this quantity.
Each of these quantities can be estimated by plugging in the estimates of these variances that come out of the model fitting.
Regarding your second question - as you can see here, each ${\rm ICC}$ has a fairly clear interpretation. I would argue that the interaction ${\rm ICC}$ does tell us something interesting - how "similar" are measurements that share both subject and site?
One important point to note is that if subjects are nested within sites, then the Subject ${\rm ICC}$ is not meaningful in it's own right, since it's impossible to share Subject and not site. Then $\sigma^{2}_{\eta}$ becomes only a measure of how much more similar individuals are to themselves, compared to other individuals at their site.
