From the upper bound $\sqrt{2}n$ on the edge length of the smallest regular simplex containing the unit cube in $\mathbb{R}^n$, shortly outlined in the question Smallest regular simplex containing the unit cube in $R^n$, it follows that roughly $n^n2^{n/2}$ unit simplices are sufficient to cover the unit cube by translation.
This might be far from optimal, since the translation covering density of any convex body in $\mathbb{R}^n$ is only $n\ln{n} + n\ln\ln{n} + 5n$, due to Rogers. Moreover, it is known that the most economical covering cannot be lattice like. (For the details and references, please let me refer to Section 1.3 in P. Brass, W. Moser, J. Pach, Research problems in discrete geometry, 2005.)