It implies a loss of composure and an inability to think or act clearly.
Here are some examples:
* "I was beside myself with joy when I heard the news." (Overwhelmed with happiness)
* "He was beside himself with rage after the accident." (Overwhelmed with anger)
* "She was beside herself with grief after her mother's death." (Overwhelmed with sadness)
Essentially, it suggests that the person is so emotionally charged that they feel disconnected from their usual self.