How will the mountains bear Shalom?
Rashi and Radak (from Shocher Tov): When they make Peros 1 , people [are satiated, and] are not stingy; "one will call to his colleague [to join him] under his vine and fig tree" (Zecharyah 3:10). Radak - when Peros are scarce, if Ploni finds his friend in Ploni's field, he quarrels with him. Since [the king] will judge Your nation b'Tzedek (verse 2), there will be satiation in the world - "Im b'Chukosai Telechu; v'Nasati Gishmeichem b'Itam" (Vayikra 26:3-4).
Radak: This is a metaphor for Nochri kings that will give Shalom to Yisrael in his days. Also "Shim'u Harim Es Riv Hashem" (Michah 6:2) refers to kings and Gedolim. So it was in the days of Shlomo, and so it will be in the days of Mashi'ach; they will be with Yisrael in Tzedakah and straightness.
Malbim: There will be much grain in the mountains - "Fisas Bar ba'Aretz b'Rosh Harim" (verse 16).
How is this connected to the king?
Refer to 72:3:1:3.
Malbim: Constancy of Tzedek and Mishpat in the nation depends on (a) the judge - "he will judge Your nation b'Tzedek" (verse 2); (b) the nation - sometimes they veer Mishpat due to their poverty, and engage in theft and fraud. Since there will be much grain (refer to 72:3:1:1*), there will be Shalom and no quarrels. Sometimes they veer Mishpat because they are Resha'im without Yir'as Hashem; refer to 72:3:3:2.
What is the meaning of "u'Geva'os bi'Tzdakah"?
Rashi: The hills will bear Shalom for them via the Tzedakah that they will do.
Malbim: They will all serve Hashem and fear Him.