Lauren S. answered 10/10/19
The first step to a good education is a passion for learning.
In this passage the lawyer agrees to stay ten extra years to prove a point. The preceding discussion by the characters concerned the death penalty and whether serving a life sentence would be preferable. The lawyer states that the life sentence would be preferable, but the banker calls his bluff, saying that he couldn't stand five years in prison.
The decision by the lawyer to raise the stakes is meant to prove his point that a life sentence would be preferable to a death sentence. Since the character is a lawyer, it is implied that this character has experience with the legal system and would likely have wanted to match more closely the actual length of time a prisoner serving a life sentence would have to serve. You could also infer that, since his chosen profession is being a lawyer, he would likely be very abstinent and eager to prove a point past what is necessary just to solidify his argument.
On the larger picture, the play The Bet mostly shows the ways in which the lawyer improves himself during his stay in prison, ultimately becoming a better man by the end of it. The main effect of the increased time is to show that "material goods" and "fleeting things" are less important than "divine salvation" and bettering oneself. Shortly before this play was written Chekhov had taken a trip to the Ukraine Steppe to recover from illness and overwork, and it's likely that this trip is largely what inspired the main idea of this play.