How exactly does capsaicin cause apoptosis in cancer cells?

Possibly many ways. This is the active ingredient in chili peppers that causes a burning sensation. It binds to the vanilloid receptor subtype 1 (vr1), which triggers a temperature oriented inflammatory reaction. Some of its effects on cancer are inconclusive, like stomach cancer. It appears to have proposed effects of other cancers, like breast cancer by causing apoptosis by affecting the egfr/her-2 pathway.
Complex biochemistry. One theory is that capsaicin induces apoptosis by generating reactive oxygen species and disrupting mitochondrial transmembrane potential. Also "a possible mechanism of capsaicin-induced apoptosis is the activation of caspase 3, a major apoptosis-executing enzyme. " [see cellular molecular biogy letters 2009 by yang and colleagues].