Why would Pelishtim rejoice?
Rashi: They aggrandized themselves in the days of Achaz. His evil caused him to be handed into their hands (Divrei ha'Yamim II, 28:18).
Radak, Malbim (28): They rejoiced when Uziyah Melech Yehudah died. He fought Pelishtim, and broke the walls of Gas, Yavneh and Ashdod, and built cities in Ashdod and Pelishtim (Divrei ha'Yamim II, 26:6). After he died, in the days of Achaz, Pelishtim rebelled and conquered a large part of Yehudah (ibid., 28:18), until Achaz died. The Navi tells them not to rejoice, for now Chizkiyah will reign, and subdue you a second time. He is from the seed of the one who struck you before.
Malbim: Refer to 14:29:2:3.
What is the meaning of "Nishbar Shevet Makech"?
Rashi: Malchus Beis David was weakened and lowered. They used to strike you; we find that David and Uziyah did so (refer to 14:29:1:3).
Radak: Uziyah [died].
Malbim: Yehudah descended wondrously in the days of Achaz, and could not overcome you.
Why does it say "Ki mi'Shoresh Nachash Yatza Tzefa"?
Rashi: From the root of that Nachash came Tzefa, which is worse than a Nachash (normal snake). "Tzefa" refers to Chizkiyah - "Hu Hikah Es ha'Pelishtim v'Es Azah v'Es Gevulah mi'Migdal Notzerim Ad Ir Mivtzar" (Melachim II, 18:8).
Radak: Uziyah is compared to a snake that strikes and kills.
Targum Yonasan: From the sons of the sons of Yishai will come Mashi'ach. Radak - we find that Yishai is called Nachash 1 - "Asher Ba El Avigal Bas Nachash" (Shmuel II, 17:25).
Malbim: Shoresh Nachash are the early kings of Yehudah. Tzefa damages more.
Shabbos 55b: This teaches that he died only due to [the snake. (Death was decreed on all people via Chet Eitz ha'Da'as.)
What is the meaning of "[Tzefa] u'Firyo Saraf Me'ofef"?
Rashi citing Targum Yonasan: They will subjugate you like a [fertile] snake.
Radak: This refers to Chizkiyah. He will be harsher for you than Uziyah, just like Tzefa and Saraf are harsher than a [normal] snake. "Me'ofef" means that it jumps from place to place. The Navi says, cease from the great Simchah that you had when Uziyah died.
Malbim: The fruits of this Tzefa will be venom that flies and damages far away, i.e. Chizkiyah and the fruits of his good deeds that will harm you.