The root cause of the death and breaking of the primordial kings — the מלכין קדמאין — is that they consisted entirely of forces of harsh judgment (דינים גדולים וגבורות חזקים) without any admixture of ḥesed or the higher air (אוירא עלאה), and Sefer HaDe'ah I (3:3:1) cites the Vilna Gaon's teaching that because the world-order began solely in the secret of the lamp of darkness — the crown of gevurah alone — it could not endure until the supernal air of ḥesed descended, after which it was sustained.
Etz Chaim (Shaar 8, Chapter 1) establishes the structural reason for the breaking: the Nikkudim emanated initially only in their circular (עיגולים) dimension, which corresponds to the level of nefesh alone, without the upright (יושר) dimension — and because no aspect of yosher emerged for them, they could not receive the supernal light and therefore broke and died, in fulfillment of the verse "God made man upright, but they sought many reckonings" (Kohelet), where "they" refers to the Nikkudim who went only in the way of circles.
Sefer HaBiurim II (8:3:32) deepens this by specifying that the kings came into being solely through the force of מ"ס (middat ha-din) without any participation of the קד"א, which is the root of ḥokhmah — and since it is written "wisdom gives life to those who possess it," existence sustained through din alone without ḥokhmah was inherently impossible; they emerged momentarily by virtue of their derivation from the root of binah, which is also called the source of life, but could not persist.
Sefer HaDe'ah II (2:2:1) draws out the consequence: the death and destruction of the primordial kings — meaning the departure of their lights and the shattering of their vessels — was itself the generative event that produced the 974 pre-creation generations, the destructive angels, the evil spirits, and all the roots of the worlds of Beri'ah, Yetzirah, and Asiyah of the world of Tohu, so that the breaking was not merely a failure but the very material from which subsequent existence was constructed.
The Zohar's Idra Rabba is the foundational source for the entire discussion: Zohar (Idra Rabba 3:128a) describes how before the Ancient of Ancients arranged His configurations, kings were engraved and sketched but their configurations did not endure — alluded to by the verse listing the kings who reigned in the land of Edom — and they were laid aside and hidden away until a later time, the land of Edom being the place where all judgments are bound together and from which no world could stand.
Etz Chaim (Shaar 9, Chapter 1) traces the cascading structural effect: each king's breaking caused a partial descent of the posterior aspect (אחוריים) of Abba and Imma, who had been sustaining the kings face-to-face; once there was no one to raise the feminine waters and maintain that face-to-face orientation, Abba and Imma reverted to back-to-back, and by the time all seven vessels had shattered, even the posterior aspects of Yisrael Saba and Tevunah had descended — the breaking of the Tohu-world thus directly reconfiguring the structure of the worlds that followed.