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This expression is not the clearest way of writing this, but the idea is that we are defining the components of the covector $Q(q)$ on a basis of differential 1-forms $dq_j$, i.e. $Q = \sum_{j=1}^k Q_j dq_j$. This is made more clear by [Wikipedia's expression of this statement](https://en.wikipedia.org/wiki/Generalized_forces#Generalized_forces): $Q_j = \sum_{i=1}^n \langle \mathbf F_i, \frac{\partial \mathbf r_i}{\partial q_j}\rangle$. We get the result from the book via: $$\begin{align}
\sum_{j=1}^k Q_j dq_j & = \sum_{j=1}^k \sum_{i=1}^n \left\langle \mathbf F_i, \frac{\partial \mathbf r_i}{\partial q_j}\right\rangle dq_j \\\\
& = \sum_{i=1}^n \sum_{j=1}^k \left\langle \mathbf F_i, \frac{\partial \mathbf r_i}{\partial q_j}\right\rangle dq_j \\\\
& = \sum_{i=1}^n \left\langle \mathbf F_i, \sum_{j=1}^k \frac{\partial \mathbf r_i}{\partial q_j} dq_j\right\rangle \\\\
& = \sum_{i=1}^n \left\langle \mathbf F_i, d\mathbf r_i\right\rangle
\end{align}$$
where the final equality uses the [differential calculus expression of a differential 1-form](https://en.wikipedia.org/wiki/Differential_form#Differential_calculus) in terms of basis 1-forms: $d\mathbf f = \sum_{j=1}^k \frac{\partial \mathbf f}{\partial x_j}dx_j$. It's a little confusing but not incorrect to explicitly write out the components on one side but incorporate them on the other. This is compounded by the fact that the differential 1-form $d\mathbf r_i$, among other relevant concepts, is never explicitly defined in the book (at least prior to here). It doesn't need to give a ground up reconstruction of differential manifolds, but it probably wouldn't have hurt for it to have spent an introductory section or at least an appendix setting notation and terminology for it.