Intuition behind the max-min inequality

I think it makes sense when you step through the proof heuristically. So using your analogy, imagine first walking across the $z$ axis and for every $z$ finding the value of $w$ that minimizes $f(z,w)$. Now clearly this is going to be smaller or equal to any $f(z,w)$ for all $z$ and $w$ because we essentially just did that operation and picked the smallest value. Now suppose we take that list of $w$ that gives us the minimum value for any $z$ and we now only consider the value of $z$ that gives us the largest value of $f(z,w)$ for that list of minimizing $w$ values. This is the max-min value. And note that because we already knew that no matter the choice of $z$ or $w$ this would be less than or equal to the corresponding $f(z,w)$ we have the inequality,

$$\sup_z\inf_w f(z,w)\leq \sup_z f(z,w)$$

Now we are almost done. All that is left is to realize is that this inequality holds for any choice of $w$ on the right-hand side and so it must also hold for the minimizing $w$ of $\sup_z f(z,w)$. And so,

$$\sup_z\inf_w f(z,w)\leq \inf_w\sup_z f(z,w)$$

So what this is saying is that when we found the lowest valley for every $z$, even when we then found the $z$ so that this valley is the "highest lowest valley". It has to be as low or lower than the "lowest highest peak" because we know the highest lowest valley is lower than every peak from the first expression. But then it must also be lower or at least as low as the lowest peak.