mixed model variance-covariance matrix| parameter estimation

First, it helps to consider what dimension the matrix $V_i^{-1}$ needs to have so that the given matrix products are defined. You'll find that it needs to be a square matrix of dimension $n_i times n_i$, where $n_i$ is the length of the vector $y_i$ (as well as the number of rows of $X_i$). In the book's nomenclature, $y_i$ is the vector of observations in some group $i$. Therefore, it becomes clear that $V_i$ has to be the VCV-matrix of responses in group $i$, $y_i$. (Side note: the estimator in (13.28) is of the form $(X'W^{-1}X')^{-1}X'W^{-1}y$, which would be the Generalized Least Squares (GLS) estimator for $beta$, if the response $y$ were known to have the VCV-matrix $W$. It is not exactly the GLS estimator, because the VCV-matrix does actually depend on an unknown parameter $theta$ which has to be estimated.)

This might already help you figure out what matrix V should look like! But the answer is actually also in the book.

Spoiler alert: from (13.25) and (13.26) of the book, you see that $$V_i = sigma^2(Z_iDZ_i' + R_i),$$ where $Z_i$ is the matrix of regressors for the random effects of group $i$, $D$ is the VCV matrix of random effects, $R_i$ is the correlation matrix of the residuals in group $i$, and $sigma^2$ is the residual variance. You'll obtain the whole matrix $V$ as a block diagonal matrix with the matrices $V_i$ on the diagonal.