We study the problem of explainable clustering in the setting first formalized by Dasgupta, Frost, Moshkovitz, and Rashtchian (ICML 2020). A $k$-clustering is said to be explainable if it is given by a decision tree where each internal node splits data points with a threshold cut in a single dimension (feature), and each of the $k$ leaves corresponds to a cluster. We give an algorithm that outputs an explainable clustering that loses at most a factor of $O(\log^2 k)$ compared to an optimal (not necessarily explainable) clustering for the $k$-medians objective, and a factor of $O(k \log^2 k)$ for the $k$-means objective. This improves over the previous best upper bounds of $O(k)$ and $O(k^2)$, respectively, and nearly matches the previous $Ω(\log k)$ lower bound for $k$-medians and our new $Ω(k)$ lower bound for $k$-means. The algorithm is remarkably simple. In particular, given an initial not necessarily explainable clustering in $\mathbb{R}^d$, it is oblivious to the data points and runs in time $O(dk \log^2 k)$, independent of the number of data points $n$. Our upper and lower bounds also generalize to objectives given by higher $\ell_p$-norms.