I'm not 100% sure how this works. But it's possible it forms a tetrahedral intermediate that is trapped with a negative charge on oxygen. Then in acid work up it collapses to make the ketone. This would also explain the selectivity for only forming ketone and not alcohol with a second addition of grignard reagent.
NMe2 anion can be a leaving group in extreme cases, though, during the elimination of anionic tetrahedral intermediates.
Careful you are not proposing Sn2 at an sp2 carbon, by the way. I'm pretty sure you aren't, but I wanted to be clear on that.