Sun Masters

It is Fohat who guides the transfer of the principles from one planet to the other, from one star to another--child-star. When a planet dies, its informing principles are transferred to a laya or sleeping centre, with potential but latent energy in it, which is thus awakened into life and begins to form itself into a new sidereal body.
- H.P. Blavatsky, The Secret Doctrine, Volume 1, Stanza VI

So to put it briefly, "as above, so below." The same process of life and death that happens to us has its corresponding analog in the cosmos.

And with regards to your second question, my understanding of this topic is that it is not the case that ordinary planets "evolve" into suns. Due to the way that the laws of nature work (and note that I am not just referring to physical laws here), it is virtually impossible for a planet to spontaneously ignite into a sun. That's not how it works.

Rather, a Monad's body as a planet or a sun is a reflection of the level of light in that Monad. When a Monad reaches a level of development that corresponds to a sun, the Monad gains the right to inhabit the body of a sun in its next incarnation.

Again, "as above, so below." The physical body that we have is largely a result of previous causes that are playing out. Even if we alter those causes, because they have already crystallized physically, the effects do not automatically and spontaneously change. Degenerating to the level of an animal in this life does not instantly transform our body into that of an animal. Similarly, reaching a level of development commensurate with a higher level of existence does not instantly render us an existence in the corresponding body, because the complexity and density of the physical world means that the effects of our causes take a long time to be carried out.

However, the next body that we have can incorporate the causes we set into motion in this life. Thus, the same applies with planets. As above, so below.