How do you explain this?

[Mireya]

So Vegeta in his laybed stated Freeza destroyed the planet and almost all the Saiyajins since he feared the surging of a SSJ... Yet he let Vegeta live, when this latter was the most likely one to become a SSJ?

[baconisbacc]

Thanks to some good luck, Vegeta was off planet when the Extermination occurred. Frieza realized that Vegeta could be one of his most powerful lackeys. Of course since Frieza despises the Saiyans, he would have killed Vegeta, sooner or later. If Vegeta never came to Earth and all that stuff never happened, Frieza would have probably killed Vegeta anyways, he just let him live for longer so as to serve his purpose.

[Yuji]

Freeza was afraid of the Saiyans not only because of the Super Saiyan legend but because they could team up and defeat him. Handling one (well, three) Saiyan by his lonesome would guarantee Vegeta's growth would be stunted under his supervision.

[Lionel]

I like what the others have mentioned. Vegeta was a unique individual whom Freeza seemed to harbour a peculiar sentimental attachment for; possessive and exploitative so it's not in the least positive in nature but it's an attachment none the less. Why else would he have made such a bold proposal to Vegeta during the ROF arc if there wasn't an interest in the Saiyan Prince? Freeza could apparently tolerate Vegeta so long as he didn't step out of line and kept to his allotted role in the PTO.

[ayakuweb]

Everyone's replies are great.

I also think that Vegeta (and I guess Nappa and Raditz) was just a natural at conquering planets so Freeza kept him in his employ.

[theherodjl]

Freeza did believe in the legend but he didn't really know much about a Super Saiyan itself. The last SSJ to appear was a thousand years earlier and barely any details about that Saiyan lasted the test of time. Freeza probably assumed that letting a few Saiyans live wasn't too much trouble as the odds of a a SSJ appearing was greatly diminished with only 3 left.

[Koitsukai]

Freeza always had Vegeta covered, he even had Namek Goku (the strongest saiyan ever at that time, using a a godly technique that multiplied his power) covered...

I guess, Freeza believes the saying: keep your friends close but your enemies closer.

[blacksymbiote]

Simply put, Frieza never had a reason to worry about a few saiyans he knew to be no match against him. He mentions thinking the only saiyan who might be able to become a SSJ would be Vegeta, but his level was nowhere near enough. Frieza was more concerned about a mass saiyan uprising when he killed the others.

[miguelnuva1]

Vegeta was the strongest saiyan ever and he was stuck at 18,000. Vegeta likely would have been assassinated had he reached a level to rival the Ginyu Force let alone Frieza.

[nickzambuto]

Freeza feared the whole saiyan race rising up. One lone saiyan, no matter how talented, is not a threat. Even if Vegeta's power level started growing exponentially, Freeza would probably have eliminated him as soon as he surpassed Freeza's first form.

[BWri]

You can't forget that Vegeta was an absolute peon before the insane power jumps on Namek. There was nothing to indicate that he would be any trouble to Frieza at all. Vegeta's clash with Goku on Earth changed everything and put Vegeta on a path of the Super Saiyan. Before that, he may have naturally died not even reaching Captain Ginyu's level of power. If he ever got to Ginyu's power level, he'd likely find himself on Frieza's kill list. I'm not sure Frieza would care prior to that.

[Jack Bz]

He didn't kill Vegeta because Beerus didn't tell him to

[BWri]

He didn't kill Vegeta because Beerus didn't tell him to

Oof, low blow.