Hello Durran,
Interesting question.
The short answer for this is buoyancy.
Long answer:
When the flask and the balloon setup is placed, it is inside a fluid (air) medium. And as it occupies a certain volume, the setup would experience a buoyant force. (If the density of the whole setup is less (hypothetically) than that of the air, it will float way, of course. But in reality, the density is high, and so it sinks.) Now when the baking soda and vinegar in the flask reacts, the gas inflates the balloon. This increases the volume of the whole system, which makes it less dense compared to the previous state. As the density decreases, the buoyant force on it increases a bit, making it to weigh lighter than before.
The law of conservation of mass is not violated here as the total mass remains the same, while the weight (mass × acceleration due to gravity) of the system is the one that changes.
Hope it helps! :)
