I also never understood the relevance of the d'Alembert principle from a modern point of view. It's just historically earlier than the Hamilton action principle, which nowadays is the fundamental starting point of all dynamical theories. It's an extension of naive Newtonian mechanics in so far as it allows to incorporate holonomous and anholonomous constraints as does of course the action principle. Given the fundamental lemma of variational calculus the principles are pretty much equivalent:
The Hamilton principle defines the dynamics by the stationarity of the action
$$A[q]=\int_{t_1}^{t_2} \mathrm{d} t L(q,\dot{q},t),$$
where ##q## are ##f## generalized coordinates. If there are constraints, they have to be incorporated as restrictions on the allowed variations, i.e.,
$$\delta q^j f_j^{(\alpha)}(q,t)=0,$$
where ##j## runs from ##1,\ldots,f## (Einstein summation implied), and ##\alpha## labels the constraints, ##\alpha \in \{1,\ldots,k \}##, ##k<f##.
Then the constraints can be worked in with help of Lagrange parameters, leading to
$$\delta A[q]=\int_{t_1}^{t_2} \mathrm{d} t \left (\frac{\partial L}{\partial q^j} -\frac{\mathrm{d}}{\mathrm{d} t} \frac{\partial L}{\partial \dot{q}^j} + \sum_{\alpha} \lambda^{\alpha} f_j^{(\alpha)} \right ) \delta q^j \stackrel{!}{=}0,$$
and since this should be true for all ##\delta q^j##, you can leave out the integral, and what you get is d'Alembert's principle.
