Would you say it's accurate to say LIGO is a big interferometer measuring changes in lengths of the arms due to gravitational waves, with some extra analysis to figure why it still works even though the light waves get stretched? Or is OP correct that such an explanation means I "don't have even a slight understanding of how it works" and a correct explanation must "involve higher spatial dimensions (not time)?"
I'd say that's a perfectly fine explanation. And in fact the light does get stretched. For sufficiently low frequencies, that's fine because there's enough time for new (unstretched) light to enter and exit the apparatus. At higher frequencies this causes the sensitivity of LIGO to fall.
Part of the popular confusion around how LIGO works is the freedom in coordinates: there are different, perfectly good definitions of space and time you can use, and the explanation sounds different in each one. So people can get them mixed up. For example, my previous paragraph makes sense in "transverse traceless gauge", but not in others.
I'm not sure what GP was referring to with "higher spatial dimensions".
