Here is a reference to a more general result: Lipschitz functions over a doubling metric space (rather than $[0,1]^d$). The $||\cdot||_\infty$ $\epsilon$-metric entropy of such functions is, disregarding log factors, of order $(D/\epsilon)^{ddim}$, where $D$ and $ddim$ are the diameter and doubling dimension of the metric space, respectively. A modern proof may be found in Lemma 4.2 here: https://www.sciencedirect.com/science/article/pii/S0304397515009469

Here is a reference to a more general result: Lipschitz functions over a doubling metric space (rather than $[0,1]^d$). The $||\cdot||_\infty$ $\epsilon$-metric entropy of such functions is, disregarding log factors, of order $(D/\epsilon)^{ddim}$, where $D$ and $ddim$ are the diameter and doubling dimension of the metric space, respectively. A modern proof may be found in Lemma 4.2 here: https://www.sciencedirect.com/science/article/pii/S0304397515009469

Edit: As pointed out in the comments, the result above is stated without proof. However, a proof is given in Lemma 6 here: https://ieeexplore.ieee.org/document/7944658/