Defining polynomial hierarchy with oracle machines and quantifiers
The oracles do not play the role of quantifiers, they play the role of everything but the first quantifiers.

First, observe how NP relates to $\exists yR(x,y)$ with $R$ in P. That means that NP allows existential nondeterminism followed by computation in P. 

Similarly, $\Pi_1$ "is" $\forall yR(x,y)$ with $R$ in P. 
Now for $\Sigma_2$ we have the same with an additional $\exists$ at the front: $\exists y_1\forall y_2R(x,y_1,y_2)$ with $R$ in P. Or, in other words, 
$\exists y_1 R'(x,y_1)$ with $R'$ in $\Pi_1$. So we have existential nondeterminism followed by computation in $\Pi_1$. And that is more or less the same as NP with a $\Pi_1$ oracle.

In general, the first quantifier specifies whether we are in the existential or universal case. The rest of the formula then describes a lower level of the hierarchy which we use as an oracle.