What are the three most important details from the story that support the theme?
This depends upon which theme we are focusing on. One of the major themes of the story is the role experience plays in life and the appreciation of life. The bet arises from a debate on which is worse: death or solitary confinement. The lawyer claims that some life (confinement) is better than no life at all.
The banker claims - even though he has no experience of being executed or confined - that capital punishment is more humane. This is a significant detail because the banker is making a claim based on no experience. This is also the subject of the bet, at least for the lawyer. The lawyer must be able to survive for fifteen years without the experience of human companionship. The lawyer claims to know that he will be able to survive, mentally intact, and then collect his reward.
The lawyer's experience during those fifteen years is relegated to books and his own thoughts. He reads voraciously but sporadically, as if searching for any meaning to justify his existence. The lawyer claims to have experienced all the world has to offer - only through books. This is a significant detail because he implies that these fleeting experiences are as fleeting as the experiences he would have in the outside world. One could argue that having no experience with the outside world and other people caused him to reject life altogether. Therefore, without that interactive experience, life is meaningless. In this case, the lawyer loses the bet; death would have been better than solitary confinement because all he got out of it was a hate for the transience of life.
When the lawyer dismisses his reward and the bet, the banker is full of contempt for himself. But we don't really know why for sure. Is he sad that he took fifteen years of this man's life, guilty for having considered killing him, etc.? It is an ambiguous ending, another significant detail, especially considering that the banker kept the lawyer's renunciation in order to protect what was left of his money. (He seemed to feel remorse but still did not want to lose any money.) Then one might conclude that the banker is only temporarily affected by his own contempt. In this sense, he confirms the lawyer's sentiments but in an ironic twist; it is the transience of experience that makes life bearable. Each sad event passes (unlike the prolonged solitary confinement); we get over it. Each happy moment passes; we long for the next. Life is so much better with variety, something one cannot have in prison.