Why does this grammar derive into $\beta \alpha ^*$ instead of $\alpha ^* \beta$?
Try some derivations:


$A\Rightarrow \beta$
$A\Rightarrow A\alpha\Rightarrow \beta \alpha$
$A\Rightarrow A\alpha\Rightarrow A\alpha\alpha\Rightarrow \beta \alpha\alpha$


The pattern is clear: starting from $A$, we'll generate strings of the form $\beta\alpha\dotsc\alpha$, in other words, $\beta\alpha^*$. Since at any stage we have $A$ on the left of the sentential form, we'll eventually generate strings with $\beta$ on the left.

If the grammar had been $A\rightarrow \alpha A\mid \beta$, then we'd derive strings of the form $\alpha^*\beta$.