Basically: Wittig reaction swaps out a C=O fro a C=C.
If you ever see a question asking "How do we convert a ketone/aldehyle to an alkene?" Look for the ylide.
Anyway, you asked "how"
Notice that the ylide has 3 phenyl group and a phosphorus attached to a CH2 group. If you look up the structure you'll actually see it's a zwitterion with a negatively charged CH2 group. So ylide = carbanion= nucleophile.
So the ylide will act as a nucleophile and attack the electrophilic carbonyl carbon. That's how the new carbon-carbon bond is formed that yields the formation of an alkene.
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