version. But there is more to say about this idea in the context of the poem and some of Wordsworth's other work. Wordsworth also implies that the adult can learn from his younger version of himself. When he was young, he was awestruck at beauty in nature (such as a rainbow). As an older man, he is still awestruck. He has learned to appreciate these things. He hopes that he will continue to experience nature in this way, to continue to experience nature as he did, as he learned, as a child: 

So be it when I shall grow old,
       Or let me die! 
The Child is father of the Man;
And I could wish my days to be
Bound each to each by natural piety. 

Just as he wants to continue to revere (piety) the beauty of nature as he grows older, he also wants his future self to revere his prior selves' experiences. In other words, he wants his future days to be bound to the same reverence of nature that he experienced in his prior days. Note that wording that the child is "father of man" (not "the" father of man). 

In "Lines Composed a Few Miles Above Tintern Abbey," the speaker (Wordsworth) expresses a similar, although somewhat different, sentiment. He realizes that he will not have the exact same reaction to the beauty of nature as an older man but believes that in his wisdom he will appreciate it more: 

                                    And I have felt
A presence that disturbs me with the joy
Of elevated thoughts; a sense sublime
Of something far more deeply interfused, (93-96)

Although he will experience things differently as he grows older, he thinks and hopes that his future experiences will be as profound, albeit different, as his experiences as a younger man. In much of Wordsworth's poetry, he expresses the desire to continue to see the world with the same wonder that a child sees it. 

This line of Wordsworth from his "Ode:  Intimations of Immortality," appears to be contradictory, but within the context of the poem, it is not, which, of course, is what a paradox is.  Physically, the boy cannot be the father of a man, but emotionally, and spiritually, he can. 

As he aged, Wordsworth became concerned about the loss of "secret sharers," parent-substitutes from whom and with whom a person creates.  However, he later realized that he was not deprived of his sharers, for he could draw from youth with whom he also had immortal ties:

Ye blessed Creatures, I have heard the call/Ye to each other make.../My heart is at your festival.../The fulness of your bliss.  I feel--I feel it all

Our birth is but a sleep and a forgetting/The Soul that rises with us,our life's Star/Had had elsewhere its setting...

O joy! that in our embers/Is something that doth live.../Of the eternal Silence truths that wake/To perish never...

Our Souls have sight of that immortal sea....

Thanks to the human heart by which we live.

In his ode, Wordsworth realizes that Youth can be inspiration to Age, and through Youth, man acquires the immortality of his ideas and truths.

Wordsworth begins his poem by stating how his "heart leaps up when I behold/A rainbow in the sky", and how it did this when he was a child too, and it will when he is old.  So what he means by the quote "The Child is father of the Man" is simply that as a child, he felt the same awe and respect for nature that he does as a man, and how the child, in its simplicity and innocence, is often wiser than the man, and can teach the man a few things, like a father.

A major theme of a lot of Worsdworth's poetry is the beauty of childhood-he thinks that "Heaven lies about us in our infancy!", that we are closer to wisdom and God when a child than when an adult.  As an adult, we don't still have that awe and reverence that children do, "The things which I have seen I now can see no more" and the world ruins us.  So Wordsworth feels that the child, having more wisdom and being closer to God, can be the greater teacher, or father, to the man.  So the paradox (a statement of something that is seemingly contradictory or impossible) exists because a child cannot be a literal father to a man, but, Wordsworth is saying that he is, in wisdom, perspective, spirituality and groundedness.

I provided links to another Wordsworth poem that follows this theme also ("Ode:  Intimations of Immortality").

Is so interesting to see how many interpretations we all have about his phrase, which is what makes his poetry so magical and intense as an experience. I feel that, as some have posted, Wordsworth is saying that our heart speaks for our brains, in not such exacting words. If we are children at heart, our inner child will dictate all the great and the wonderful things that we find in life. If we aren't born with an inner child, our life will lack that, and we might even lose control of it.

In addition to the answer above, this line could also be religious in nature, due to "Child" and "Father" being capitalized.  We can think of this line as the child is Jesus, the father is God, and man is everyone on Earth, in one interpretation.  Also, we can look at it like this: the only way to salvation is through Jesus, according to Christian beliefs, because Jesus was sent to bear all of our sins through his suffering and death.  Jesus, the child of God, was the father of men because he came onto this Earth, preached and shared his beliefs about salvation and about God, and died so that his "children" could be saved, much as a father would do if one of his children was in danger of dying or being killed.

In his famous ode to nature, William Wordsworth says that the child in every person teaches him to appreciate nature beginning with the simple beauty of rainbows and by implication, other natural wonders. What we think as children will help determine how we think as adults. The lines that follow "the child is the father of the man", suggest, with almost religious zeal, that he hopes to always love nature as he did as a child.

Wordsworth included the final three lines of "My Heart Leaps Up" as an epigraph to his "Ode: Intimations of Immortality from Recollections of Early Childhood." In the "Ode," Wordsworth celebrates his experience with nature and life in his youth. He also laments losing the wonder he felt during these experiences, feeling that as he grew older, he'd lost some of that wonder: 

Whither is fled the visionary gleam?
Where is it now, the glory and the dream? 

However, by the end of the poem, Wordsworth finds reasons to remain hopeful, that he might continue to find profound significance in the experience of ordinary things: "To me the meanest flower that blows can give / Thoughts that do often lie too deep for tears." As an adult, he might be less enchanted with a rainbow (or a flower) but he is wiser and therefore can experience more profound meaning in such experiences. 

Wordsworth intends a similar gesture in "My Heart Leaps Up." He does not go into such detail as he does in the "Ode", but he does indicate that he will continue to be emotionally moved by natural beauty, such as a rainbow, at all stages of his life.

Comparing the "Ode" and "My Heart Leaps Up," consider the idea that in Wordsworth's life, his childhood occurred before his adulthood. In a linear sense, the Wordsworth child existed before the Wordsworth adult, as if to say that the child lived before the adult and therefore the child's experiences taught the adult how to experience things. As an adult, Wordsworth recalls his childish wonder and learns from his childhood experiences. Just as a son feels loyal to his father, the adult Wordsworth feels loyal to himself as a child and he feels loyal to his future self and future experiences. This is what he means when he says that he wishes his days (his life and life stages) to be bound to each other by "natural piety" (natural duty). 

Another interpretation of this phrase is that of the son (the child) becoming a father to the man (the father), as roles sometimes becomes reversed with aged parents.

Wordsworth's poem speaks to the power of childhood in the formation of one's identity.  The poem articulates a condition of universal subjectivity.  The vision of " a rainbow in the sky" is the natural element that unifies the speaker's, presumably Wordsworth's, sense of identity.  The beauty of the rainbow, "so was it when my life began; So is it now I am a man; So be it when I shall grow old," is timeless.  It serves as a way to connect the fragmentation to one's consciousness that is the result of age.

It is here in which Wordsworth suggests that there is much to be learned from childhood.  "The child is the father of the man" is his way of articulating how childhood provides the basis for the growth and maturation of the individual.  Wordsworth suggests that a significant way for an individual to understand their own condition in the world is to examine their own childhood.  The experiences that Wordsworth has as a child has played formative roles in developing his identity as an older man.  For Wordsworth, the man grows from the child within the individual.  In the child being "the father of the man," Wordsworth suggests that the innocence of childhood and the moments that help forge who we are as children play formative and vital roles in the life we lead when we become older.

The child that he was predicts the man he will become: just as it did as a child, the natural beauty and magic of a rainbow still has the power to make his heart leap. Despite the knowledge and understanding that comes with maturity (he now knows something of what makes a rainbow presumably), the beauty still moves him.

MY heart leaps up when I behold  
  A rainbow in the sky:  
So was it when my life began,  
  So is it now I am a man,  
So be it when I shall grow old         
    Or let me die!  
The child is father of the man:  
And I could wish my days to be  
Bound each to each by natural piety.

When the speaker sees the rainbow, his heart "leaps up" with joy. He was happy to see rainbows as a baby, is still happy to see them as an adult, and hopes - one day - to be happy to see them as an old man.

Thus, his childhood enjoyment of the rainbow connects his childhood to his adulthood - and hopefully, will connect his adulthood to his old age. The rainbow and the pleasure the speaker takes in it is a constant point in his life: an unchanging pleasure in a changing world.

It is because his childhood enjoyment of the rainbow connects him as a child and an adult: hence, the man he is grew from the child he was. For that reason, the child might be thought to be father of the man.

Wordsworth is illustrating the ways in which the attitudes, feelings, and actions of adults are based on the attitudes, feelings, and actions of those same individuals as children. He says that he reacted to the sight of a rainbow in a certain way - his heart leapt at the beauty - "when my life began" and that, years later, he still reacts in the same way "now (when) I am a man." Furthermore, he expects that the same reaction will still be present "when I shall grow old."

Yes, that phrase is the theme of the poem. Childhood becomes the foundation for adulthood, preceding the later situations and shaping the patterns of later years in the same way that a father shapes and forms the actions taken by his child.