This passage occurs on Christ the King Sunday, Year A, most recently November 2017.
Much like the beatitudes, it is hard to preach this text without steering off the cliff of works righteousness. A few thoughts. First, a goat and sheep are born that way; the sheep did not become sheep by their actions; neither for the goats. They are declared righteous, but the text never declares them righteous because of their action. It simply says they are righteous. They did X, Y and Z good things. Lutherans believe the righteous do good things. Second, the sheep are not endeavoring to save their hides but they are simply helping people. The goats were perfectly willing to help Jesus to help themselves, but they weren't interested if it didn't get them points. The whole freedom in faith righteousness is that we no longer have to work about our own reputation (glory) or status before God but instead can worry about our neighbor. The goats never got that far. Lastly, for Matthew glory is found in judgment. For Lutherans we believe that judgment comes on the cross, which points toward the cross being the center of glory. Even if this seems stretching it the basic point of this text is a theology of the cross: Jesus's glory is revealed, yet still somewhat hidden, in the brokenness of the world.
δοξα ("glory"; 25.31) It is interesting to note that in the Gospel of Matthew the word δοξα is almost always connected with Jesus second coming and judgment. Perhaps it is worth reflecting on -- what is so glorious about judging?
εθνος ("gentiles" or "nations"; 25.32). When used in the plural it normally means "gentiles" ie, non-Jews. Jesus will finally tell us to go to all the nations.
κληρονομήσατε (from "κληρονομεω" meaning "inherit"; 25.34) This word can mean receive, but it really involves inherit. An inheritance means two things: First, that someone died. Second, that there is a gift. The kingdom given to us is a gift in Jesus Christ and his death.
ξενος ("stranger"; 25.35) The phrase, 'I was a stranger and you welcomed me' actually has the word: "xenos" as in xenophobia. Furthermore, the verb is "synagagete," from which we get synagogue. To translate a different way: "I was an outsider and you gathered me to worship." "Synag-oo" as a verb does not mean invite to church, but the word underneath means gather. I think Jesus is implying something stronger than simply welcoming strangers but more like: ushering in freaks.
Grammar: Unclear antecedents
Like in English, Greek uses pronouns. Sometimes it is unclear what "it" is refering to. For example, the Greek says, "throne of glory of his." Is the throne his or the glory? Probably doesn't matter in this case, but worth reminding ourselves that Greek does have ambiguities.
In 25.32 the object of the word "divide" is interesting. Jesus has just finished talking about the εθνος (gentiles), which is a neuter noun. The pronoun object of the word divide is a masculine plural, suggesting the nations are not what are divided, but the individuals in the nations (masculine plural pronouns can refer to a group that has both men and women).