A forge is the place where the blacksmith makes a very hot fire. He then uses the fire to soften metal so he can bend and hammer it into a new shape. Therefore, in the poem, the flaming forge is his fire, which has flames just like a fire in a fireplace does. It is the place where he does his life's work.
In this first use of "flaming forge" in the poem, the children of the village love to look in the door of his workspace.
Later, this literal forge is likened to "the flaming forge of life." In this instance, the word "forge" is a turned into a metaphor for the hard work we must do to achieve whatever it is we are meant to accomplish in life. The blacksmith working at his forge to shape his metal is compared to how we must shape or forge our own lives. The blacksmith with his steady, hardworking, responsible ways, is a good role model for all of us in living fruitfully and meaningfully. His work is humble, but he does it faithfully and well.
The awe of the children shows that the blacksmith is a part of the life of the community and that his work is valuable. Though he is such a powerfully muscled man, with strong arms, the children are not afraid to watch him at work—keeping a distance, of course, by standing in the doorway so that they don't get hurt.