How do you write a letter to explain why a child was absent so many days from school?

"How" doesn't really matter much. You can either justify it (the child was in the hospital) or you can't (nearly anything else, really... I suppose "kidnapping" would be another valid reason). It's not like the school is going to say "so obviously this parent kept the kid out for trivial reasons, but it's such a beautifully Worded letter explaining it; let's drop the whole issue."