I cannot understand the short verse or poetry in the beginning of the book. What is it about ? Why does he start with this?

Alice Liddell asked Carroll to write the story he told that summer day. It took him approximately two years to do so.  He delivered the book to Alice on Christmas day.

The poem in the begginning of the book describes the circumstances of the day the story was created. Carroll and Duckworth were rowing the boat down the river. The three sisters asked Carroll to tell them a story.  The oldest daughter,Lorina ("prima"), told him "to begin it." The middle child, Alice ("secunda"), demanded "there be nonse in it!" Edith, the youngest ("tertia") kept interrupting while the story was being told "not more than once a minute."

Carroll created a wonderland through which a "dream-child" wandered and her experiences there. He tired of telling the story, telling the girls he would finish it next time they were together.  They insisted that he continue, thereby creating the "tale of Wonderland."  They returned home.

Alice Liddell was favored by Lewis Carroll. which is why she became the main character of the story.  In the last stanza of the story he tells her to take the story as he wrote it in memory of the day. He wrote it down as she requested, and presented it in memorium to the beautiful day the had so long ago.

***The relationship between Lewis Carroll and the Liddell's had been cut off at some point during the time it took Mr. Carroll to deliver the book. This may account for the sadness apparent in the last stanza.