In the second part to this question, the solution uses the product rule to express the partial derivative of f with respect to y in another form.
Why is this necessary and how is it possible? What context is this done in ie. is there any specific topic I should go back and learn to understand this step?
Because you mention the product rule I'm assuming you are talking about the line that says:
$$ \frac{\partial^2f}{\partial s \partial t} = \frac{\partial}{\partial s} \left ( s \frac{\partial f}{\partial y} \right ) = \frac{\partial f}{\partial y} + s \frac{\partial}{\partial s} \left (\frac{\partial f}{\partial y} \right ) $$
This isn't expressing the partial of $f$ with respect to $y$ in a different form, as you mentioned in your question. It is evaluating the partial derivative with respect to $s$ of the expression $ s \frac{\partial f}{\partial y}$. Because $f$ is implicitly a function of $s$, this means that $ \frac{\partial f}{\partial y}$ could also depend on $s$. Thus, you are taking the derivative of a product of functions of $s$, so you need the product rule. Then, the final equality on the right is just what you get from applying the product rule.