Your quoted theorem (actually, a generalization where the second set is $[0,n)$ or even $[0,\infty)$) appears as Lemme 32, p. 267 of the [Banach-Tarski paper (1924)](http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-fmv6i1p27bwm).

This is on the way to their third main result (see Introduction and Thm 35, p. 270): In $\mathbf R^n$ ($n\geqslant 1$) any two subsets $A$, $B$ with nonempty interior are "equivalent by countable decomposition", i.e. $A=\coprod E_i$ and $B=\coprod g_i(E_i)$ for some disjoint $E_i$ and displacements $g_i$.

The only antecedent they note is by [Sierpiński (1918, p. 142)](https://babel.hathitrust.org/cgi/pt?id=umn.31951d00697922m;view=1up;seq=154): "Let us remark that, by using Mr. Zermelo's axiom, one could decompose a square into countably many sets with which one could then compose (by a suitable translation of each of these sets) a square larger than the given one."