Brain waves during REM sleep are the same as when awake — it's the stage where you dream.
Literal
REM-sleep-during [possessive-の] brain-waves [topic-は] awake-being time [with-と] same brain-waves are, dreams [object-を] see stage is.
であり is the formal copula's connective form, linking two facts about REM sleep. ~ている時と同じ compares the brain-wave state: 'the same as when (you are) awake.' The medical-loanword ステージ (stage) sits naturally alongside native medical vocabulary like 脳波 (brain waves).