Here's why:
* Broly's Power: Broly, as a baby, was already incredibly strong. His immense power threatened King Vegeta's position and the stability of Planet Vegeta.
* Fear of Broly: King Vegeta feared that Broly's uncontrolled power would eventually pose a threat to his rule and the Saiyan race.
* Broly's Cry: When Broly's power was awakened due to his father's angry cries, King Vegeta was horrified. He saw Broly as a dangerous liability.
* Political Motivations: King Vegeta, being a ruthless and ambitious ruler, likely wanted to ensure the safety of his own power and the Saiyan race. He saw Broly as an uncontrollable force that could disrupt his plans.
In later iterations of the Broly story:
* Broly in Dragon Ball Super: The Broly seen in Dragon Ball Super is a different interpretation of the character. In this version, Broly was banished by King Vegeta due to his power and potential threat. This is in contrast to the original movie, where Broly was targeted for execution.
* Broly's Father's Fate: King Vegeta ordered Broly's father to be executed as well, because of his uncontrolled anger and the potential for his son to follow suit.
So, while King Vegeta didn't directly kill Broly in the original movie, he was the driving force behind his execution, fearing his uncontrollable power and the threat it posed to his rule.