As I wrote elsewhere: if any newer fans are confused by the video, here's an explanation.
The video seems to open with umbilical cords, knotted together, before they morph into a fetus. Then we see an unmistakable image of maternity: a mother's breasts, and her arms around her child. The synchronicity beween lyric and video is perfect, as the line "please stay a child" overlays exactly with the graphics, before "somewhere in your heart" matches up with a scribbled, red heart.
The band members' heads appear next, as if to signal paternalism: U2, collectively, is the father of this child. As Bono has said, this song is for "all of U2's girls." The female child in the video also represents the future, the coming generation of music, and the result of U2's influence on the world.
As the video continues, Bono whispers in the child's ear, "I'll give you everything that you want..." as the child smiles, listening. The advice continues, as written phrases appear on the child's head, scrawled like words of wisdom in a classmate's yearbook ("Stay True"..."Let Go"). Like a father imprinting life lessons into a child's mind, Bono's lyrics implore listeners to recognize their own individuality.
The flowers growing from the child's mouth symbolize her individuality, her own song, and her identity. Bono instructs her to "shout it" everywhere she goes, and the flowers represent the child's unique voice.
Bono has said that while the song was first written for Edge's daughter, in the end, he "eroticized it" a little bit. The video hints at this fact, as the child transforms into a woman, emerging from the silky cocoon of childhood to enter womanhood. The director's imagery is telling: we see butterflies (a symbol of post-pubescence), a pregnant womb, and a rose (the quintessential symbol of romantic, sexual love).
Even the tattoos on Bono's arm are symbolic: we see a heart, "Mother," and a dove flying free, which echo the themes of the song.
As the song climaxes, Bono's face teeters on the edge of joy (of a father seeing his daughter grow up) and agony (the ache of a father missing his little girl).
There are so many themes in this song: parental influence vs. individuality, nature vs. nurture vs. free will, the entry into adulthood, the excitement it presents to a teenager, mixed with the sadness a father feels from seeing his daughter leave home.