>>40165578sex hormones literally work by binding to hormone receptors in the cell's nucleus. This triggers hormone-related gene transcription in the cell. If it's an androgen then that makes the corresponding masculine gene, and an estrogen makes a corresponding feminine gene, and this alters cell activity.
A cell that develops in a hormonally male environment becomes structurally and actively male. A cell that develops in a hormonally female environment becomes structurally and actively female. This can happen in a male or female body because everyone has a full genome that codes for either a male or female body, aside from the few testicle related genes on the Y chromosome.
If you're born male and start taking estrogen, your new cells become structurally and actively female, and over time as your old cells die, the new female cells develop and eventually organize into female tissues. That's how breast growth and skin changes work.