"The child is the father of the man" is one of Wordsworth's most famous quotes and beliefs. The idea of childhood is an essential component to Worsworthian poetry and thought. On one hand, Wordsworth sees childhood as a moment where purity and honesty can be released into the world. This period of life is one where one is fully immersed within their world and the barriers of in-authenticity and duplicitious conditions are not present. In Wordsworth's poetry, the idea of childhood is firmly linked with the expression of emotions, the reverence for the natural world, as well as the idea that social conformity is secondary to individual authenticity. The connection between both nature and childhood converge in Wordsworth's poetry.
Some aspects of this question has been tackled before in the references below. Wordsworth often selected children in the natural environment as the inspiration for his poetry; best perhaps exemplified in poem Lucy Grey. In this poem, the young child becomes lost and dies in the snow. Her spirit, however, is believed by Wordsworth to live on in the natural world.
Yet some maintain that to this day
She is a living child
That you may see sweet Lucy Gray
Upon the lonesome wild.
Wordsworth used children to illustrate the simplicity, wholesomeness and innocence of man in the natural world, and also because as characters they would use 'the language really used by men' -simple but rich diction which Wordsworth admired and respected above the purple prose of earlier poets.
William Wordsworth seems to have had a soft spot in his heart for both Nature and Youth. Both, he perceives have innocence. One poem by Wordsworth which illustrates this point very well is from the Lyrical Ballads 'Anecdote for Fathers.'
In this simple warm and heartfelt poem, he outlines a walk between a father and a young child in which they discuss the merits and demerits of the country homes they have lived in and which the young son prefers.
The child is refreshing in his innocent honesty, and says he prefers the old house! However, despite his tender five or so years, there is the dawning of a more mature realisation of how one human's words affect another's feelings. On seeing the father's surprise (and perhaps disappointment) the child appears to cover what he said, perhaps to sweeten the pill. He adds he only liked Kilve (the old house) better because of it's weather vane. Wordsworth may have been touched by a story of a child's tender feelings towards a father he didn't want to hurt.
The poet had many siblings himself and was separated from them a lot when their parents died during their childhood. He was also very close to his sister Dorothy with whom he shared an awe of nature.
"The Child is the Father of the Man."
The positive and life nurturing impressions that get deeply etched in our minds when we are small children remain with us for the rest of our adult life. These early spiritual life sustaining childhood memories form the foundation of our adulthood and determine our personality. Hence, a happy childhood filled with pleasant life giving positive spiritual influences -here symbolized by the pleasant associations linked to the beautiful rainbow - is the 'father' or the creator or the architect of a mature adult, the 'man.'
William Wordsworth was a Nature poet who worshipped Nature as his God and it was his main source of spirtual comfort and escape from all the cares of this world. His association with life giving and life sustaining Nature began even when he was only a child and remained with him till his death.
In this short lyric, the 'rainbow' symbolizes the life sustaining and life nourishing goodness of Nature. The sight of the beautiful rainbow which he saw when he was only a child is deeply etched in his memory and the same joy that he experienced when he saw it as a child contiunes to remain with him through his adulthood. He desires that this same childhoood joy should continue to sustain him even in his old age. Wordsworth says that he would rather die than not being able to experience the same joy that he experienced when he saw the rainbow when he was a small boy after he becomes an old man.
The memory of the beautiful rainbow and its pleasant associations form the link between his childhood, adulthood and his old age:past, present and future. Wordsworth concludes the poem by expressing the desire that each day of his existence be linked with the next by beautiful and simple natural sights like the rainbow.
For Wordsworth the life nourishing and life sustaining memories of beautiful natural sights like the rainbow are very precious and he deeply desires that they link each day of his life on this earth and remain with him till his death.