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Multitape Turing machines against single tape Turing machines

Introduction: I recently learned that a multi-tape Turing Machine $\text{TM}_k$ is no more "powerful" than a single tape Turing machine $\text{TM}$. The proof that $\text{TM}_k \equiv \text{TM}$ is ...
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Why does it take $O(f(n)^2)$ to simulate a 3-tape $O(f(n))$-time TM on a 1-tape TM?

This looks like a fundamental result but I can't find a resource online that gives an intuitive interpretation of this complexity. Any basic explanation is appreciated.