Let's begin by looking at the passage:
Matthew 6:1-4 Take heed that ye do not your alms before men, to be seen of them: otherwise ye have no reward of your Father which is in heaven. Therefore when thou doest thine alms, do not sound a trumpet before thee, as the hypocrites do in the synagogues and in the streets, that they may have glory of men. Verily I say unto you, They have their reward. But when thou doest alms, let not thy left hand know what thy right hand doeth: That thine alms may be in secret: and thy Father which seeth in secret himself shall reward thee openly.
Matthew, chapters 5-7, contains what is known as the Sermon on the Mount. Matthew 6:1-21 deals with the importance of laying up treasures in heaven instead of piling them up on earth. One of the major ways we accomplish this is by doing our good deeds quietly and not as a show to be seen by others. Jesus gives three examples of this: the giving of alms (v.1-4), prayer (v.5-15), and fasting (v.16-18). Those who do these good deeds in a showy way in order to be seen of men have already received their reward--the praise of men. However, they will receive no additional heavenly reward for such actions. However, those who do these things secretly (that is, without a big show) have laid up treasures in heaven (v.19-21).
The giving of alms is the giving of money to help the poor and needy. No one should give alms to be seen of men (v.1); they should not sound a trumpet announcing their good deed (v.2); but should do the deed in secret in order to receive God's reward (v.4). Verse three and its teaching about not allowing the left hand to know what the right hand is doing is part of this argument.
When we do something good, we naturally feel good about ourselves. We tend to want to tell someone else so they can tell us how good we are. Jesus tells us that we should not be telling others of our good deeds. But if we keep rehearsing in our minds what a good thing we have done, we will tend to slip and tell others of the deed. The best way to keep from "spilling the beans" is to avoid thinking about it ourselves; to avoid making our actions a part of our inner conversation. This, I believe, is what Jesus is speaking of here.
Our right hand and our left hand are both a part of us. When one of them lets the other know what it is doing, it is carrying on an inner conversation; a conversation in the brain. Jesus is telling us: Don't even think about it; don't congratulate yourself for your good deed; don't think about how few people would have done such a thing; don't dwell on the act. Even self-praise can lessen your reward. And, the more you think about it, the more likely you are to tell someone else. Don't even let your left hand know what your right hand is doing.