"But how does a change in the law in New York State change this, except for people in New York State?

Right now, it doesn't. It is merely a step... but an important one. It would take perhaps a Constitutional amendment to ensure ALL states followed the same rules.... and that would take many years to get it ratified.

...Now, because New York doesn't require residence to be married, many would go there and hope that their state would recognize the marriage. It would at least be a psychological victory.

This United "States" arrangement is a complex one...it has advantages...and many disadvantages. Marriage laws, voting laws, liquor laws, economic laws...and more... can all vary. It allows states like Delaware and South Dakota to be financial 'home' to credit card companies...and states like Kansas to practically eliminate legal abortion.

You can barely imagine how deeply the idea of some states not wanting to be bound by the laws & practices of others is ingrained here. The South is the prime example, but Nevada gets to host Las Vegas ...etc.

Advantages? Well, this way some states CAN legalize same-sex marriage when it might not be possibly to do it nationally for many years.