When Starkiller Base is:
destroyed, as everyone is flying away from it, it appears to turn into a planet-sized star itself.
Is that what happens, or did something else happen,
like the whole surface turning molten hot or something?
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According to the novelization, the entire planet imploded, leaving a binary system, meaning that the destruction created another star or a similar planetary body.
In moments they were beyond the atmosphere of the imploding planet. The jump to lightspeed was accomplished without incident, preventing them from observing the final cataclysm. Which was just as well.
A moment after they fled, Starkiller Base system became a binary.
When describing the results of the Starkiller's attack on the Hosnian System, the novelization uses a similar description.
In its wake, the detonation left behind a blazing, spherical mass. The home of the Republic had become a new binary system: one utterly devoid of life.
The weapon on Starkiller Base was not just powered by stars, but seemingly used the mass of a star as a projectile. While the weapon was charging it took in the matter and energy of its star (or a neighbouring star?) The movie didn't make this very clear, but I assume that the weapon had some kind of containment field to store the matter/energy until it was fired. So when the Resistance attacked the planet the facilities supporting the containment field were destroyed, and the stolen star mass expanded to form a star where the planet had been.
According to Wookieepedia, it exploded when the Resistance attacked it, due to a weak point that was damaged by Poe Dameron. This would indicate total destruction, not turning into a star.
Starkiller Base was an ice planet that was located in the Starkiller Base system of the Unknown Regions that served as the headquarters of the First Order three decades after the Battle of Endor. The base was commanded by General Hux and was armed with a superweapon capable of destroying entire star systems, which the First Order viewed as an evolution of the planet-destroying capabilities of the Galactic Empire's Death Star designs. It was destroyed by a Resistance attack, due to a weak point which was damaged by pilot Poe Dameron, causing the explosion of the whole planet.
Although I must admit that I thought the same when I saw the movie, there isn't any factual evidence that anything happened after the attack that triggered the massive chemical reactions needed to create a star.
We must assume, in absence of other evidence, that it was just a decorative explossion.