I understand the solution to part a) it's the second part i'm having trouble with. I understand that $\frac{dv}{dt}=0$ implies that the distance from the the fixed point remains constant, however I do not understand the conclusion "solutions stay on the level sets of V". Any tips on plotting these levels sets would be useful also.
No. The function $V$ is not the distance from fixed point.
A solution is a parametric curve $x=x(t)$, $y=y(t)$. Some computation with chain rule leads to $$\frac{d}{dt}V(x(t),y(t))=0$$ This means, the composition $V(x(t),y(t))$ is constant, independent of $t$. Let $c$ denote this constant. Saying that $V(x(t),y(t)) = c$ for all $t$ means precisely that the parametric curve $x=x(t)$, $y=y(t)$ is contained in the level set $\{V=c\}$.
As for sketching, I would not want to do that by hand. But if you must, note that $V=c$ can be solved for $y$ in terms of $x$ and $c$, using quadratic formula. Since $V$ cannot be negative, use $c=0$, $c=1$, $c=2$... hopefully that will be enough for a sketch.