Why does the author say "memory" when he means "disk space"? Or does he actually mean "memory", as in "RAM"? In that case, why would the size of the executable matter?
As for why the size of the executable matters, you needed to have the entire executable in RAM before you could actually execute it. That's not so true these days, but it was back in the good ol' days* when you hooked tape cassettes up as your external storage.
* Get off my lawn!
The size of the executable mattered because it all had to fit in 1024 bytes RAM (no "K", no "M", certainly no "G") with all video & operating system memory.
By the way, to whomever downvoted a legitimate question: Nice.
In some sense, it's not a legitimate question for this context.
One reply puts it into historical context, and educates the reader about the memory/persistence model of the ZX81.
The other reply points out that the lack of virtual memory and whatnot means that, in previous times, you needed to have the entire thing loaded into memory to execute something.
Without some digging and a knowledge of what to look for, those are non-obvious but useful points.
First thing that appears is a Wikipedia article which explains how the memory and off-line storage worked.
So I don't think finding this out would have been rocket science.
Having said that, and having written my first code on a ZX81 (if you don't count an even older TI programmable calculator with even less RAM) I'm old enough to be amused by the question. :)
However you slice it, chess in 1K was awesome coding.
punishing a person for being ignorant while they are trying to learn and become less ignorant is terrible behavior.