The Dunn article was published in Science, which means that most of the details are in the supplemental materials. Here's the relevant part:

Just before receiving their money, participants were asked to complete the Positive and Negative Affect Schedule (PANAS; S16), as well as reporting their happiness on the same single-item measure used in the previous studies. That evening, after spending their windfall, participants again completed the PANAS and a modified version of the single-item happiness measure (specifically, participants were asked to rate their overall happiness that day on a 5-point scale anchored with the words “not happy at all” to “extremely happy”). We standardized the 10 positive affect items of the PANAS and the single-item happiness measure to create reliable 11-item indices of happiness both pre-windfall (α = .81) and post-windfall (α = .87). A preliminary ANOVA on prewindfall happiness revealed no between-group differences, [F’s < 1], enabling the use of ANCOVA (with pre-windfall happiness as a covariate) in the main analyses of postwindfall happiness.

So happiness was measured with 11 items, 1 directly asking about happiness and 10 asking about positive emotions. Each item was rescaled so that the average of all subjects on that item was 0 and the standard deviation was 1. Then the 11 items were averaged together. Those who were instructed to give to others scored .37 points higher on that composite happiness measure at the end of the day than the spend on self group, controlling for scores on that composite happiness measure at the beginning of the day. Since the SD of that composite measure was about .64, that means that they were about .6 SD's happier, which is generally considered a "medium" effect size.