root-n consistent estimator, but root-n doesn’t converge?

What hejseb means is that $\sqrt{n}(\hat\theta-\theta)$ is “bounded in probability”, loosely speaking that the probability that $\sqrt{n}(\hat\theta-\theta)$ takes on “extreme” values is “small”.

Now, $\sqrt{n}$ evidently diverges to infinity. If the product of $\sqrt{n}$ and $(\hat\theta-\theta)$ is bounded, that must mean that $(\hat\theta-\theta)$ goes to zero in probability, formally $\hat\theta-\theta=o_p(1)$, and in particular at rate $1/\sqrt{n}$ if the product is to be bounded. Formally,
$$\hat\theta-\theta=O_p(n^{-1/2})$$
$\hat\theta-\theta=o_p(1)$ is just another way of saying we have consistency – the error “vanishes” as $n\to\infty$. Note that $\hat\theta-\theta=O_p(1)$ would not be enough (see the comments) for consistency, as that would only mean that the error $\hat\theta-\theta$ is bounded, but not that it goes to zero.