The Savior himself is prophesying that the Gentiles would reject the fulness of the gospel. In this context, this means that they will reject the Book of Mormon and the gospel message carried by the missionaries. This event is very important for it marks when the times of the Gentiles is fulfilled, when the times of the Gentiles is come in, a light shall break forth among them that sit in darkness, and it shall be the fulness of my gospel; But they receive it not; for they perceive not the light, and they turn their hearts from me because of the precepts of men. And in that generation shall the times of the Gentiles be fulfilled (DC 45:28-30).
The phrase "the first shall be last and the last shall be first" is applicable. In the meridian of time, the Savior declared that the gospel was to go first to those of the House of Israel and second to the Gentiles. In the last days, the order is reversed, the fulness of the gospel is to go to the Gentiles first, and secondly to the House of Israel. Given that many of the House of Israel (particularly of Lamanite and Nephite descent) have already turned to the Lord, we can see that the times of the Gentiles is coming to a quick close. Furthermore, the scourge spoken of in DC 45:31 is already upon us, demonstrating that we are the generation in which the times of the Gentiles is to be fulfilled.
Joseph Fielding Smith
"'TIMES OF GENTILES' TO BE FULFILLED. We go unto them with a message of peace, of truth, of eternal salvation, calling upon them to repent of their sins and enter into the true fold, where they may receive rest. When they will not do this, but to the contrary, will listen to the unrighteous and condemn the truth, then God will withdraw the gospel from among them. In that day the times of the Gentiles will be fulfilled and the gospel will be carried to the Jews. For, this gospel must be preached to them as well as to the Gentile nations; and a remnant of the Jews will gather-as they are gathering since the dedication of their land for their return-in Palestine, and as a remnant of Ephraim and his fellows are now gathering to the land of Zion." (Doctrines of Salvation, vol. 3, p. 8)