Why does the LIGO observation disprove higher dimensions?

I’m the lead author of the paper. Thanks for being interested in the work! Your question is a good one. Really, our work can’t say anything about extra spatial dimensions if they’re not doing anything to gravity or light. As you correctly mention, we can only constrain higher dimensions where gravity is actually leaking into them.

If there are higher dimensions, but our physics experiments can’t see or hear them, are they really there? :p (this isn’t to say there might not be other ways of detecting extra spatial dimensions — but really, if they aren’t affecting physics in any measurable way, there’s not much we can say)

It doesn't disprove all possibilities for higher dimensions - technically, you can't really disprove something so broad because there's always another way to phrase it that will put it out of reach of existing experimental data. This is a common theme with science, and thus why that scientific claims and hypotheses have to be specific and stated precisely.

What it does do is, as you've surmised, disprove the idea, or at least the idea specifically tested, that gravity "leaks" into higher dimensions as a specific explanation for why it is so weak. This is in turn posited by some conceptions of string theory, but not all.

I suspect the reason you are thinking that it somehow disproves higher dimensions generally and thus are confused when you see that it actually doesn't, is because of bad media. The media is not being anywhere close to as precise with these claims as a scientist would or should be, and is touting it as having "disproved higher dimensions", not "disproved a particular gravitational theory that says gravity weakens through leakage into higher-dimensional spaces".