t-butyl chloride is insoluble in water, that is, when you added the NaOH solution you will get two immiscible phases. The reaction, therefore, happens at the interface of the two phases, making it slow as the OH has to get to the t-butyl chloride to cause the elimination. Adding an amount of acetone will probably cause the phases to partially mix resulting in a faster reaction. You didn't say how much acetone you used, if you used enough you may only get one phase in that case the reaction will be rapid.
So it is a solvent effect causing partial dissolution of the two immiscible layers resulting in a faster reaction.