Soy isoflavones are similar (but not identical) in structure to the hormone estrogen. This structure allows them to bind to estrogen receptors in the body, resulting in either estrogenic or antiestrogenic effects, depending on where they bind.[1] In particular, soy isoflavones bind to estrogen receptor beta (ERβ) more readily than to estrogen receptor alpha (ERα). ERβ is involved in programmed cell death (apoptosis) and cell differentiation and is thought to oppose the cell proliferation effects of ERα (in other words, soy isoflavones can function as ERα antagonists).[2] This is particularly relevant when it comes to hormone-related cancers; see Can soy isoflavones be used to prevent or treat breast cancer?.
Other proposed mechanisms that are independent of estrogen include tyrosine kinase inhibition and antioxidant activity.[1] Tyrosine kinase is important for cell growth and proliferation, and so inhibiting it could contribute to anticancer effects, while antioxidant activity could reduce inflammation.