[1][2][3] The reaction achieves results similar to the addition of an allyl Grignard reagent to the carbonyl.
The reaction involves electrophilic allyl shift via a beta-silyl carbocationic intermediate, the beta-silicon effect.
In the given reaction, the electrophilic compound (carbon with a ketone group) is treated with titanium tetrachloride, a strong Lewis acid and allyltrimethylsilane.
According to the general principle, the Lewis acid first activates the electrophilic carbon in presence of allyltrimethylsilane which then undergoes nucleophilic attack from electrons on the allylic silane.
Secondary carbocations are high in energy, however it is stabilized by the silicon substituent ("β-silicon effect", a form of silicon-hyperconjugation).