It's not money (or material possessions), it's freedom. Freedom to pursue your ideas, freedom to have options, freedom to do your own thing, freedom to give, or not to waste time on things you hate.
All these things can definitely make a person happier.
Money can be the limiting constraint on having more options or freedom. Now the study only shows a correlation with between log-income and happiness but it's not crazy to assume that having health insurance, a savings account, money for flexible travel, all increase your freedom and thus happiness.
All these things can definitely make a person happier.