In this question, there's an informal answer that gives an intuition to the Mahalanobis distance. From my understanding, the Mahalanobis distance first transforms the vector, using the inverse of the covariance matrix $\Sigma^{-1}$, to map it into "uncorrelated" space (denoted by the circle in the linked question). If we wanted to measure the distance of two points within the "uncorrelated" space, would we not have to use the formula:
$$ (\Sigma^{-1}(x - y))^T\Sigma^{-1}(x-y)$$
in order to map the left-hand vector from the original space to the "uncorrelated" space as well? Sorry if I didn't formulate my understanding very well, I'm not a mathematician.