- Donate to causes you care about + get access to influential people to further those causes
- Receive top medical care for physical and mental health problems
- Drop everything to care for or spend time with sick friends and family
- Engage services like lawyers and accountants to make stressful problems go away
I could go on...
And sure, having money is not to going to guarantee things like good personal relationships. But it's a hell of a lot easier when you can throw money at many sources of stress.
The more wealth you have the more time you have to do things you want to do.
I wonder if this is around the transition point between being fairly well off, but still salaried and reliant on the salary, to being independently wealthy (or at least, the prospect of achieving independent wealth). Highly paid but salaried people are subject to all sorts of politics and power games, and there's always the prospect of losing it all.
Observing the people I know who are truly independently wealthy, the thing they have that no salaried person has is control of their time. Even if they're nominally employed, it tends to be in a flexible manner. That control over your life is a step change in well being.
Edit: some reviews of the app used for the experiment hint at this bias 
Like let’s say I have a great marriage, good well behaved kids, lots of close friends, so I have maybe a network of people I can turn to, I have people to help me when I need emotional support etc, and having these things not only allows me to experience well being but then makes it easier to get and keep higher paying jobs?
To be fair I think it just goes both ways, but I really hope people don’t think more income is a magic solution for peoples’ problems leading to their happiness. I think the hard work it might take for a low income earner to get a higher income, is part of what also increases their well being.
(Take everything I say with a grain of salt because I haven’t had money problems for almost my entire life)
Money can buy you free time, by reducing the time you spend working for someone else (two jobs -> one job -> no job), and by paying other people to do stuff for you. Money reduces stress from your life, by knowing you can quit your job if it becomes too bad, by knowing you can afford to fix broken stuff or get healthcare (or get better healthcare). Money allows you to buy things you need, and the things you want.
This can improves your relations, because you have more time to spend with people you care about, more time to think about them, can spend money to help them, can spend money to have fun with them. Being less stressed makes it more fun to be with you. Having money makes you a more desirable partner (other things being equal).
And this all can translate into better income opportunities. Being happily married is cheaper than being divorced. Having "f-you money" improves your negotiating position at job. Being less stress makes it easier to focus on being productive. You can buy the tools or lessons that you think you need; you have time to learn new stuff. Your friends can recommend you better jobs, or better business opportunities. (Also, rich people usually have rich friends, which usually have access to better jobs and better opportunities.)
This cycle can also break if you e.g. have some mental problem or personality trait that makes your life suck regardless of how much money you have.
> I think the hard work it might take for a low income earner to get a higher income, is part of what also increases their well being.
It can make you more proud of your achievements. Depends on whether you would rather be proud, or live an enjoyable life... speaking for myself, I would choose the latter, but maybe the grass is always greener on the other side. Also, your hard work may fail to make you rich, or you may get older and unable to work hard anymore.
Even so, the fact that it's logarithmic rather than a cutoff says much the same thing: beyond a certain point it takes a lot more money to produce a little additional happiness. Tradeoffs for other factors (hobbies, finding a partner, social connections, doing public good) might produce more additional happiness than trying to increase your salary 10x.