We separate the membrane and all other cell components, but how is it that only plasmid DNA is isolated and not chromosomal DNA?
It comes down to the size of genomic DNA (relatively huge) compared to that of plasmid DNA (rather small).
Cell lysis for plasmid extraction involves an alkaline buffer that includes sodium hydroxide and SDS. Both plasmid and genomic DNA denature in this condition. When the alkaline reaction is neutralized with potassium acetate, the fairly short plasmid DNA can quickly re-anneal, while the much longer and more complex genomic DNA tends to become tangled up with itself, which prevents sister strands from properly aligning and often results in protein-DNA binding interactions.
DNA bound to protein will then get pulled down during centrifugation, while soluble plasmid DNA remains in solution.
Qiagen and BiteSize Bio both offer some more description of the process: https://www.qiagen.com/us/resources/technologies/plasmid-resource-center/key%20steps%20in%20the%20plasmid%20purification%20protocols/