I've seen many explanations that just state they are equivalent straight away however I don't understand why they're equivalent. As far as I understand, the axiom of choice states that for any indexed family of non empty sets, it is possible to choose one element from each set.
But what does this have to do with the Banach-Tarski paradox? In one of my books on measure it says that if you assume the Axiom of Choice, then there is a subset $F$ of the unit sphere $S^2$ in $\mathbb{R}^3$ such that for $k \in [3,\infty)$, $S^2$ is the disjoint union of $k$ exact copies of $F$: $$S^2 = \bigcup_{i=1}^{k}\tau_i^{(k)}F$$ where each $\tau_i^{(k)}$ is a rotation. This then apparently leads to the conclusion that the 'area' of $F$ has to simultaneously equate to many different values.
But at no point in this process do I understand where the axiom of choice is employed. Nor do I understand why $F$ has to take different values - once you've chosen the subset $F$ at the start then surely it's fixed and $F$ is just how big you chose it originally? Why does it have to have simultaneously many values?
Banach Tarski is prooved by picking a representative from an infinite set (orbit equivalence classes). This requires the axiom of choice. However banach tarski does not imply the axiom of choice, they are not equivalent.