Why can 2 uncorrelated random variables be dependent?

Correlation is a measure of linear dependence, so it kind of gives you an indication of how the two variables are related linearly. It doesn't capture however more complicated behaviour.

Therefore if you have $X$ and $X^2$ with $X \sim N(0,1)$, then

$$\operatorname{Cov}(X, X^2) = E(X^3) - E(X)E(X^2) = 0$$

but the two random variables are clearly dependent.

No, it doesn't imply that. Correlation is just a one-dimensional measure, whereas dependence can take many forms. For instance, the indicator variable of the event that a normally distributed random variable is within one standard deviation of the mean is uncorrelated with the random variable itself, but is clearly not independent of it.