What do the ∀ and ∃ symbols mean in the Axiom of Choice?
$\forall$ reads as "for all", and $\exists$ reads as "there exists".
So, in english we have

$$\text{"if }\underbrace{\text{for all $x$}}_{\forall x^\sigma}\text{ }\underbrace{\text{exists a $y$}}_{\exists y^\tau}\text{ with $R(x,y)$, }\underbrace{\text{then}}_\to\text{ }\underbrace{\text{there is a function $f$}}_{\exists f^{\sigma\to\tau}}\text{ so that }\underbrace{\text{for all $x$}}_{\forall x^\sigma}\text{ holds $R(x,f(x))$".}$$

I skipped over the $\sigma$ and $\tau$ superscripts, as they indicate types and are not of primary importance here.