The only change from Matthew 5:38 is the now familiar change to an emphasis on “it is written,” rather than “it hath been said.”
These two verses deal with the way in which we are to retaliate against those who might harm us. The “eye for an eye” rule creates social balance by requiring that what one inflicts on another person might be inflicted against the perpetrator. This is a particularly horrible inversion of the Golden Rule.
The amplification for the rule of retaliation was that one should obey the law willingly. In the Old World, Jesus’s listeners were subject to Roman law. Even if they desired it, the “eye for an eye” rule was impossible in their situation. Even the ability to resist might be removed. Hence, it was wise not to retaliate (and make matters worse). The solution was to accept the subjugation that they could do nothing about.
Nevertheless, this was not an abject acceptance of being belittled. In order for one to be struck on the right cheek by a right-handed person, it would have to be a backslap, which was highly insulting. The typical result would be an escalation, and attempt to regain honor by some form of retaliation. Particularly under Roman rule, that could be disastrous.
However, by turning the other cheek, the person could symbolically both recognize that they were politically powerless, but insist that the person doing the smiting treat them as an equal, with the forehand slap rather than the backhand. Jesus was teaching his listeners to retain their dignity in a difficult political situation.
The situation was not like that for those listening to Jesus, at that time. However, perhaps all of them could remember a time when they also underwent oppression. It would still have been an important lesson to learn from their past, if not in their present.