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Given that $z$ is a complex number.

In the proof that I have seen, $z = re^i\theta$ which leads to $\lim_{r \to 0} e^{3iθ}$. The statement alone shows the limit doesn't exist but I do not understand why.

The reason given is that the limit of $r$ tending to $0$ depends on the angle of approach; why does $r$ tending to $0$ have anything to do with the angle $z$ makes with the real axis? I can't understand this intuitively.

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    To keep it simple, consider approaching zero from the positive side of the real axis and also from the negative side of the real axis. Do you see the mismatch/discontinuity? That is "why" there is no limit. – 2017-02-28

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In the plane, there are lots of paths that tend to zero. This argument shows that the limit is different when you go to zero in different ways (along different rays emanating from zero), so it can't exist.

This is a generalisation of the idea for functions on the real line, where the limiting value when approaching the point from below has to agree with that when approaching from above for the limit at the point to exist.