The following pieces of your model are fixed and known: $X_i$ and $Z_i$. The vector $\beta$ is fixed but unknown. You have two random pieces in your model, $b_i$ (I would expect this to be $b$, i.e. shared for all $i$, and I'm going to use $b$ below), and $\epsilon_i$. I suspect the setup of your problem also specifies that $b$ is independent of the $\epsilon_i$.
To see where the variance expression comes from, just plug your model formula in:
\begin{align}
\text{var}(Y_i|X_i) &= \text{var}(X_i\beta + Z_ib + \epsilon_i)\\
\text{(a)}&=\text{var}(Z_ib + \epsilon_i)\\
\text{(b)}&=\text{var}(Z_ib) + \text{var}(\epsilon_i)\\
\text{(c)}&=Z_iDZ_i^T + R_i(\gamma),
\end{align}
where (a) is because $X_i\beta$ is constant, (b) is because $b$ and $\epsilon_i$ are independent, and (c) comes from looking at the distributions of $b$ and $\epsilon_i$.
Putting the "among-individual" and "within-individual" language into context is problem-dependent. For example, mixed-effect models can be used to represent penalized splines for scatterplot smoothing, in which case there are no individuals at all: the random effects are coefficients of basis functions.
For a (totally fictional) example where the language makes sense, suppose an experimenter is modeling sodium concentration $Y_i$ in pond water, and they are using potassium concentration as a covariate $z_i$. They sample 6 casks (say, 1 gallon each) of water from different locations in the pond, and from each cask, test 5 small 1 milliliter subsamples. We have $i=1, \dots, 30$ samples, and each sample belongs to some cask $k(i) \in \{1, \dots, 6\}$.
We might model this as a mean, a linear part depending on potassium, a random adjustment to the intercept depending on the cask, and a random measurement error. We could write
$$Y_i = [1\,\,\,z_i]^T\beta + Z_i^Tb + \epsilon_i,$$
where $Z_i$ is a vector of zeros except for a one at entry $k(i)$: it picks out the $k(i)$ entry of $b$. We assume $b \sim N(0,\sigma_b^2 I)$ and $\epsilon_i \sim N(0,\sigma_\epsilon^2)$. Our random vector $b$ has six entries: one for each cask, and they represent the variability between casks. You'll find that $\text{var}(Y_i) = \sigma_b^2 + \sigma_\epsilon^2$, and that $\text{cov}(Y_i,Y_j) = \sigma_b^2$ if observations $i$ and $j$ came from the same cask, but $\text{cov}(Y_i,Y_j) = 0$ if they came from different casks.
Y_i = .... I think you'll find (at least) two variances in the setup: one or more for the random effects (this is the "among-individual" source of variance), and one for the error variance (this is the "within-individual" source of variance). $\endgroup$