Question about an interpretation of the MRS
Writing out the expression with Leibniz notation, we get $\frac{\frac{du}{dx}}{\frac{du}{dy}}$, or the marginal utility of $x$ divided by the marginal utility $y$.
For example, let's imagine I get 10 utils from another apple and 5 utils from another cabbage. In that case, the MRS is therefore $\frac{\frac{10 utils}{1 apple}}{\frac{5 utils}{1 cabbage}}$, or, $\frac{2 cabbage}{1 apple}$, matching the marginal rate of substitution of 2 cabbages per apple.
A bit of simplification will easily get you $\frac{dy}{dx}$, which is the rate of change of $y$ with respect to $x$, the interpretation we're familiar with.