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Abstract: Quantum thermalization is well understood via the Eigenstate Thermalization Hypothesis (ETH). The general form of ETH, describing all the relevant correlations of matrix elements, may be derived on the basis of a `typicality' argument of invariance with respect to local rotations involving nearby energy levels. In this work, we uncover the close relation between this perspective on ETH and Free Probability theory, as applied to a thermal ensemble or an energy shell. This mathematical framework allows one to express in an unambiguous way high order connected correlation functions (here identified as free cumulants) in terms of standard correlation functions. This perspective naturally incorporates the consistency property that local functions of ETH operators also satisfy ETH. The present results open a direct connection between quantum thermalization and the mathematical structure of Free Probability, thus offering the basis for insightful analogies and new developments.