Vegeta is the Prince of the Saiyans from Universe 7, who made his debut back during the Saiyan Saga of Dragon Ball Z. He appeared a major antagonists at first, but after the Freeza Saga, he somehow joined the Z Warriors. His actions were still things that could be questioned.
Vegeta, being one of the only two pure blooded Saiyans that are relevant right now, was always going to be important to the story. Whether that was as an antagonist, or a Z Warrior. As Goku progressed, so did Vegeta. Back at Namek, Goku was able to show his superior powers to Vegeta, who before dying warned Freeza that Goku was the one who he was fearing all along. That turned out to be true, and Goku was indeed that which Freeza feared.
He was a Saiyan with the potential to become a Super Saiyan. That transformation eventually occurred when Kuririn died, and that became the cause of Freeza’s defeat. Eventually, Vegeta was able to transform himself into a Super Saiyan as well. This happened off-screen, but we did get to see how it actually happened later on.
But, the question remains, why wasn’t he able to transform into a Super Saiyan when fighting Freeza? I know it’s how Toriyama intended the story to go, so that would’ve never been possible. But when you look at it objectively, it seems weird. He was very strong when he fought Freeza. He certainly carried the weight of all the Saiyans on his shoulders that had perished, and the emotional incentive was there. The rage was there as well.
Later on, he randomly transforms into a Super Saiyan while facing some meteors, and I’m pretty sure Freeza was a much bigger threat than they were. Why didn’t Vegeta transform into a Super Saiyan at Namek? The only possible explanation to that would be lack of S-cells, because he didn’t have a kind heart. So how exactly was he able to transform into a Super Saiyan later?
Well, the key to that would be Bulma. Because of Bulma, Vegeta certainly had a change of heart. Sure, it may not be obvious, and he may not show it on surface, but Bulma changed his heart. A kind heart means more S-cells, which when combined with a situation similar to the one at Namek, gave Vegeta the Super Saiyan form.