First off, these aren't just for kids. A good poem has universal appeal, and if it captures a child's imagination it ought to capture an adult's as well. The main difference between "poems for kids" and "poems for adults" is that the former are easier to understand and enjoy at face value.
T.S. Eliot's The Wasteland goes over heads because it's full of allusions and analogies only the well-read are likely to get, whereas Where the Sidewalk Ends by Shel Silverstein is self-contained (though there's plenty more going on beneath the surface for older readers to appreciate). And who can argue that John Donne is easier to understand than A.A. Milne?
Of course, "at face value" doesn't mean there are no hidden meanings in children's poetry. It just means kids don't have to pick them up to be impacted by the poem. At the same time, teaching them to do so from a young age with the poetry they love will give them a huge advantage when they're older and encounter Milton, Frost, and Yeats.
Poetry intended for young readers is often marked by what some consider a juvenile rhyme scheme. Yet organizing thoughts and words in a metrical pattern and making it seem effortless isn't the work of some "kiddie writer," but of a master.
Kindling in children a love of poetry (not just rhymes) is essential for helping them reach their full imaginative potential. Stories are important, but poems impart a sense of wonder and an attitude of curiosity and investigation that mere prose can never accomplish. If Keats could transport grown men and women to other realms, why would we think Grahame, Carroll, or Stevenson couldn't do exactly the same thing for kids?
Readers often forget that poetry is intended to be read aloud. While children can have plenty of fun curled up on the couch with their favorite book of poems (probably mouthing each word silently as they read, deaf to the rest of the world), reading aloud with them opens their ears to the beauty of language, its cadences, its mysticism and enchantment. It'll probably do the same thing for you.
Which brings us back to the first point: poetry for children is also poetry for adults. Nowhere else is the ethos of childhood so perfectly rendered than in poems of this kind, where the happiness, terror and innocence of the young is brought to full light not just in the images and words themselves, but in the very way they're structured on the page and spoken out loud. If you feel lost or dead to the world of childhood, read some of the poetry written to and for kids before adulthood squelches any youthfulness of soul you have left.