“starve.” The Hebrew word is very rare, and only used here in the whole Bible. It most likely means something like “make lean,” and “starve” (or “cause to waste away”) is a very likely translation. Various English translations include, “starve,” “waste away,” “shrivel,” “weaken,” and “destroy.” The gods of the nations will not be worshiped; there will not be any sacrifices or sacrificial offerings burned to them, so they will starve to death. This is a graphic way of saying they will disappear or be destroyed.
“will bow down to.” Or, “will worship.” The same Hebrew verb, shachah (#07812 שָׁחָה), is translated as both “bow down” and “worship;” traditionally “worship” if God is involved and “bow down” if people are involved, but the verb and action are the same, the act of bowing down is the worship. In this context, “bow down to” seems better because some may not really “worship” even if they bow down. The common biblical way of bowing down before people or God was to fall to one’s knees and bow the upper body to the earth. [For more on bowing down, see commentary on 1 Chron. 29:20].
“each one from his own place.” There are scholars who say that “from his own place” refers to the fact that the nations will travel “from” their own place to Jerusalem to worship (cp. C. F. Keil, Keil and Delitzsch Commentary on the Old Testament). While that will certainly happen (Isa. 2:3; 60:6; Micah 4:1; Zech. 14:16; Matt. 8:11), that is not the primary emphasis of this verse. In the Millennial Kingdom people all around the whole world will worship Yahweh, and that will not only be when they travel to Jerusalem; they will worship from their own homes and lands. This will be a fulfillment of God’s promise to Abraham that “all of the clans of the earth will be blessed in you” (cp. Gen. 18:18; 22:18). God intended that the blessing of salvation and knowing Him would be available to everyone everywhere, but years after Abraham lived that blessing was obscured by the Jews, who tended to view Yahweh as their personal possession and denigrated the Gentiles. But God’s promise to Abraham did not mean that people would be blessed if they traveled to Jerusalem to worship, especially since God spoke to Abraham more than 400 years before there even was a Jewish nation and “Old Covenant,” and more than 850 years before Jerusalem was the capital of Israel and the site of the Temple.
In the Millennial Kingdom, people all over the world will worship Yahweh right at home as well as travel to Jerusalem to worship Him. For example, many years before Zephaniah lived, the prophet Isaiah said that there will be an altar to Yahweh in Egypt and a pillar erected to honor him at the border between Egypt and Israel (Isa. 19:19). Even the Assyrians and Egyptians, historical enemies, will worship Yahweh together (Isa. 19:21-25). Isaiah also said that even though there will be relatively few people left alive on earth after the Great Tribulation, people will praise and honor God all over the earth (Isa. 24:13-16). Isaiah foretold that people around the world would honor God, and because of that God would bring them to Israel and they would worship at His Temple (Isa. 56:3-8). Malachi also prophesied that Yahweh’s name would be honored with incense and offerings all over the world (Mal. 1:11).
So although Zephaniah 2:11 certainly includes the fact that in the Millennial Kingdom people from around the world will travel to Jerusalem to worship, when Christ reigns as king over the whole world people will also worship Yahweh right where they live. Jesus basically said that also, because he told the woman at the well that it would not just be in Jerusalem that people would worship, but Jesus knew from the Old Testament prophets that the holy spirit of God would be poured out on people in the Millennial Kingdom, he said that people would worship “in spirit and in truth” (John 4:20-24).
[For more on the Millennial Kingdom, see Appendix 3, “Christ’s Future Kingdom on Earth].