This explains very little. State income tax is based on your state of residence, not where you have a vacation home, in which by definition you do not live most of the year. Further, it makes no sense to refer to "the state with no income tax", as there are many of those.
Granted, there are some working-age people who buy a vacation home with the thought of moving into it permanently a decade or two into the future, but those plans entail a lot of uncertainty (health, closeness to family) and of course once they move, it is no longer a vacation home.
Alaska, Florida, Nevada, New Hampshire, South Dakota, Tennessee, Texas, Washington, and Wyoming do not have income taxes. Less than 20% isn't 'many' to me but I realize YMMV.
But when you have less money, every dollar counts. The income tax in places like Cali or NYS can be huge because it eats into the disposable income. It could easy cut your disposable income in half or maybe eat it all.
It has been my observation that no-income-tax states make up for it in sales tax and/or property tax. Government is getting its cut one way or another. Of course, retirees will want to skip CA altogether with their income tax and ridiculous sales tax.
As you sort of allude to, YMMV. I moved from California to Nevada and literally all taxation went down.
Income tax from progressive scheme nearing 10% to 0%
Sales tax from ~9% to ~8%
Property taxes are hard to pin to a single value, but with housing prices just being lower in general you wind up paying less for similar footage/amenities
While it doesn't mean the observation is fundamentally inaccurate, it's worth keeping in mind that people looking at states with 0% income tax may be coming from a place where they're just getting reamed because work wants them there. If you're not flying under the SALT cap and work is remote friendly it's worth at least looking around.
Granted, there are some working-age people who buy a vacation home with the thought of moving into it permanently a decade or two into the future, but those plans entail a lot of uncertainty (health, closeness to family) and of course once they move, it is no longer a vacation home.