Most people who have played Final Fantasy VI know that Shadow has various dreams. However, it's really easy to miss them, because the only way to see them is to sleep at an inn with Shadow in your party. Even then, they only happen randomly. They also do not necessarily happen in a linear fashion, so I am here to piece everything together! Altogether, there are five. The following consists of screencaps from the dreams and a summary of each (as well as a few inferences).
The first dream does not consist of much -- it is merely Baram calling out to Clyde for help. There is no background, and the background noises are pretty erie. However, it's our first glimpse at what Shadow looked like before he became an assassin.
The second dream is much longer and much more informative than the first. In this dream, we learn that Shadow and his friend, Baram, robbed a train for a million GP. It is after this success that Baram comes up with the name "Shadow." What's interesting is that Baram looks like a common thief (all the other thieves of the game look nearly identical), but Clyde/Shadow looks more sophisticated. Perhaps he is the brains of the operation and Baram does the dirty work? It's hard to tell.
Baram has fallen into some trouble on his own, and Clyde has come to save him. Clyde most likely find his friend because of his vision (dream #1). At any rate, Baram seems to be injured beyond the point of recovery, so he asks Clyde to finish him off so he will not have to endure any torture once he is caught (presumably, he robbed someone and narrowly escaped). Clyde refuses, and leaves Baram to suffer at the hands of those pursuing him.
I think this scene in particular is very important in understanding why Shadow became the cold-hearted killer that he did. Not being able to kill his friend, Baram, inevitably meant that he was going to be more tortured. Once Shadow realized this, he decided that he needed to toughen up. Also, it's more than likely that Baram calling him a coward got to him...
Like most of the dreams, it's unclear what happened right before. But for some reason, Shadow has stumbled upon Thamasa, sort of in a stupor. A dog rushes out (Interceptor!), and a woman follows. She asks Shadow to stay.
It is commonly thought that this woman saved Clyde, but I think it's equally likely that she knew him before. Perhaps Clyde has returned home after Baram's death, completely delirious, and his wife is begging him to finally stay and end his life of thievery. But once again, it's hard to know which interpretation is correct.
While the previous dream showed Clyde arriving in Thamasa, this dream shows Clyde leaving. He tells Interceptor to stay and take care of "the girl" (presumably Relm), but Interceptor ends up following after him. It looks like Clyde is the one who needs to be taken care of...
A lot of people ask why Clyde ended up abandoning Relm, and my only guess is that Relm's mother died and he could not stand living in Thamasa without her there. After losing his best-friend and his wife, he lost the will to live a normal life and turned to a darker lifestyle. He knew Relm was in good hands and decided that she was better off without him.