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The question is : Find the equation that passes through the point $(0,1)$, where the slope at $(x,y)$ is $xy$. I have a feeling I'm supposed to derive the answer somehow. I'm not quite sure what I'm supposed to do, I'm kind of lost here.

The answer is $y= e^{(\frac{x^2}{2})}$.

I have no idea how they got there

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    Hello. If you'd be so kind as to put the question into the body itself, and include what sorts of things you suspect you should do or might be helpful, I'd be glad to help you out. – 2017-02-08

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This is a differential equation. You have $\frac{\mathrm dy}{\mathrm dx} = xy$, which we can rewrite as $$\frac{\mathrm dy}y = x\ \mathrm dx,$$ the point being to get the $x$ terms on one side and the $y$ terms on the other. Next, integrating both sides, $$\int \frac{\mathrm dy}y = \int x\ \mathrm dx,$$ so $\ln y = \frac12 x^2 + C$. Now solve for $y$, and you can find $C$ by substituting the point $(0,1)$.