Can someone tell me the science of dreaming? What are the parts of the brain involved when we dream?

Actually, all parts of the brain are involved in someway excluding the prefrontal cortex, which normally experiences a state of low activity (transient hypofrontality).  The only exception to hypofrontality amid dream sleep is when the brain experiences a state of lucid dreaming.  Overall, dreaming is a byproduct of activations in the brain arising from the metabolic processes that occur during sleep.  Our brain is the largest consumer of our body's energy uptake and its metabolic demands persist throughout the sleep process.  Periodically, during sleep, our brain becomes active as it increases the flow of oxygen and glucose rich blood into its structure, which is stored as glycogen.  The imagery in our dreams are how our active brain, amid sleep, synthesizes the neurological effects of its increased metabolic activity. The quantitative study of dreaming is called oneirology; however, there are more important processes other than dreaming, IMO, that occur amid sleep of which many people are not aware.  One of those processes are those involving the most recently discovered glymphatic system.  I hope this helps.