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).

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 quote is by Sierpiński (1918, p. 142): "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."

**Note.** What you saw may well have been Borel's note *Les paradoxes de l'axiome du choix* (1947). By an argument which is only superficially different, he shows there that Zermelo's axiom implies "euclidean equality of the whole and the part, in a finite domain" (namely $[0,1]$), and concludes: "It seems preferable to me not to admit the axiom". He doesn't cite Banach, Tarski, or Sierpiński.