I know it's true, and understand why, but I can't see it. I feel stupid.
Perhaps I can't see it becaus too complex to hold at once, the two functions, the translation, the axes. Maybe it's not important to "see" it anyway, and I'm expecting too much... and knowing the rule and why is enough?
What If we start with function $f(x)$, then shift it vertically by $\Delta y$ and horizonally by $\Delta x$, to get another function $g(x)$, then
$$ g(x) = f(x - \Delta x) + \Delta y $$
Although we add $\Delta y$, we subtract $\Delta x$ - this subtraction is the "reverse" of the question.
Why One way to think about it is that $g$ is sampling $f$ at another input point. To get to that other point, from the point of view of $g$, we go backwards, the opposite direction. When we describe $f$ being shifted to become $g$, it is from the point of view of $f$. So, this change in point of view is why we reverse the horizontal shift. If instead, we described the shift as where we came from, it would already be "reversed".
This creates another puzzle: why isn't $\Delta y$ reversed too? Because it's a translation of the output, after the change in point of view has already occurred.
In another way, the difference between horizonal and vertical translation is an artefact of notation. They are both reversed (or, from the point of view of the new function), if notated as:
$$ g(x) - \Delta y = f(x - \Delta x)$$
Seeing It seems simpler to just follow the evaluation of the function. For $ g(x) = f(x + a) + b$, first you add $a$, then evaluate $f$ there, and finally add $b$. There's an extra layer of cognition when interpreting this as a translation of $f$ to $g$, because it entails a change in frame of reference (from $g$'s POV to $f$'s POV).
I think my confusion of this comes from how it was taught: instead of begining with function evaluation and then how it can be seen as a translation, we were taught the tramslation as a thing in itself, using the "rule" above. An "explanation" was given as an afterthought, secondary to the "rule". I'm not convinced the teacher had any real understanding beyond that, so they couldn't pass on an underetanding beyond the rule.