(I don't know Maggiore's book, but the question can be answered without, I guess.)
The notation suggests that $D_R\!\left(\theta_a\right)$ is a group element in some representation $R$, depending on a number of parameters $\theta_a$. (The index $a$ goes from $1$ to $d$, the dimension of the group, and the parameterization is chosen such that $\theta_a=0$ corresponds to the identity element.) Presumably, the representation is finite-dimensional, so you can think of $D_R$ as a matrix.
(Update: Note that $d$, the dimension of the group, i.e. the number of parameters you need to specify a group element (equivalently, the dimension of the Lie group as a manifold) is generically different from the dimension of the representation, $d_R$. A given group has infinitely many representations of various dimensions.)
Then clearly the generator $T_R^a$ is again a matrix of the same dimension, and there are $d$ independent such generators. Furthermore, the generators form a vector space (multiplication of group elements effectively turns into addition of generators), and indeed $\alpha_a T^a_R$ implies summation: This is a linear combination of generators with coefficients $\alpha_a$, forming a new generator.