The simplest way to explain the Golden Rule is that if you want to be treated well in this world, you should treat others well. If everyone follows this philosophy, everyone will treat others well and therefore, everyone will be treated well.
Another logical proposition that derives from the Golden Rule is that we are more likely to enjoy life if we contribute good deeds rather than bad actions. First, we get the social, psychological, and spiritual comfort of having done something good for other people. Second, because we contribute good things to the world, we make the world better. And since we live in the world, by contributing good things, we are directly making our experience in that world better.
We all live in the world; we all participate in life. Consider the analogy that we all drink from the same spring. If we add bad things (pollutants), we are making the water bad for everyone, including ourselves. If we add good things, we are making the water (analogously, life) better for ourselves and others.
It makes moral and logical sense to do good to and for others because we're making the world better for them and us. There is the general idea here that "everything affects everything." Think of Newton's Third Law:
For every action there is an opposite and equal reaction.
In human society, good actions tend to lead to other good actions; not the opposite evil actions. Newton was talking about force. In this context of morality, we're talking about humanity. Speaking of the Golden Rule, Newton's Law could be rephrased like this: For every good action, a subsequent good action is more likely to occur than a bad one. I not only reinforce my own good behavior; I also make others happy and therefore increase their likelihood to be good to others (including myself).