Here is the quote in context:

So when you give to the poor, do not sound a trumpet before you, as the hypocrites do in the synagogues and in the streets, so that they may be honored by men. Truly I say to you, they have their reward in full. But when you give to the poor, do not let your left hand know what your right hand is doing, so that your giving will be in secret; and your Father who sees what is done in secret will reward you.

In my Christian upbringing, this was interpreted as colorful imagery making the point that you should not let attention motivate you to do good deeds, and that in fact you should make an effort to not get attention from your good deeds by doing them in secret. The reward from 'your Father' may additionally be interpreted as that which a truly good person values - the actual goodness caused by the action.

In the study, people who have an obvious signal that they have done good feel satisfied, and thus "have their reward in full," and don't donate as much as the people with the envelope. I think the OP is implying that by intentionally avoiding signalling good deeds, we will end up doing more actual good - thus validating the advice of the scripture.