The slides say the Independence Lemma (stated below) is an "intuitive result" but I'm struggling to understand the intuition.

The Lemma states:

For \( X, Y, f(X, Y) \in L^{1}(\Omega, \mathcal{F}, \mathbb{P}) \), with \(X \space \mathcal{G}\)-measureable and \(Y\) independent of \(\mathcal{G}\), for some sub-sigma-algebra \(\mathcal{G}\). Define \(g(z)= \mathbb{E}^{\mathbb{P}}\left[f(z, Y) \right] \) for all \( z \in \mathbb{R} \). Then: $$\mathbb{E}^{\mathbb{P}}\left[f(X, Y)|\mathcal{G} \right]=g(X)$$

Any help with the intuition would be appreciated!