The carbonyl interacting with the leaving group thing is highly speculative. Ignore it for your purposes.
Basically, the transition state of the SN2 reaction is "conjugated" with the pi-orbitals of the carbonyl/vinyl group next to it. This conjugation reduces its energy, and therefore lowers the barrier to the reaction resulting in a faster reaction. Maybe drawing the five-membered transition state, complete with the p-orbitals on the neighboring group, would help?
Personally, I always just thought of it like this: having a double bond adjacent speeds up all substitution reactions by weakening the carbon-leaving group bond. The one exception is that carbonyls don't speed up SN1 reactions, because they are electron-withdrawing and you need electron-donating groups for carbocation stability.