I feel the clearest way to understand this question/answer hasn't been stated yet. Recall that the cysts in PCOS are unruptured follicles. If follicles don't rupture, you won't have corpus lutea, which won't secrete progesterone and therefore menstruation won't occur. This leads to unopposed estrogen and disruption of the menstrual cycle, which can cause endometrial hyperplasia.
The patient here is obese and has hirsutism. Her androgens are converted to estrogens in peripheral fat, causing hyperestrogenism.
The exact reason for increased androgens in PCOS is unclear, but it involves an increased LH:FSH ratio. Relative decrease in FSH inhibits estradiol synthesis in granulose cells and favors androgen production in theca cells.