Why does my child that is 11 feel dizzy?

Many reasons. "weak and dizzy" covers about everything one could suffer from, so we'll narrow it down a bit. At 11, and especially in females, eating is sometimes not so great, so low blood sugar or low blood pressure due to inadequate salt intake is possible. Children at that age do not hydrate very well, and most don't get good sleep. All these can lead to weakness and dizziness.
Vertigo. Lots of causes such as lack of sleep, visual problems , anemic , inner ear infection like labyrinthitis, hypoglycemia from not eating good. He needs to be examined.