Why does the balloon blow up when vinegar and baking soda are combined?

Actually CO2 has a density of about 1.9 g/L at STP (Standard Temperature and Pressure) while homogeneous air, including CO2 has a density of about 1.2 g/L, so actually pure CO2 would sink when trapped in a balloon.

The reason the ballon would inflate is because when you seal the balloon over the bottle, the air you trap inside is already at a pressure equal to the air outside. What happens is the CO2 released inside this closed system adding to the gas already inside. By increasing the amount of gas, the balloon is simply stretching to accomodate a change in volume, rather than a change in pressure as a cap would do. Eventually, the balloon should pop or fly off as the pressure would overcome the seal between the balloon and the bottle. Since a gas is being produce, you have to be careful not to use something that can't easily break free, otherwise, if enough gas is produced, the container or lid could break violently (explode)!

So in otherWords, simple pressure is causing the balloon to stretch in the only way it can: up! See if you can tie the balloon off without losing too much CO2, and you should find that it drops to the ground. As an additional experiment, put the balloon in the freezer, and watch what happens: the balloon will shrivel after a few minutes, even though little or no CO2 escapes. Let the balloon warm up to room temperature and it will "magically" inflate right before your eyes! the reason is because the density is decreasing, even though the "amount" of CO2 remains constant. To make up for this, the gas must expand and take up more space.

When vinegar and baking soda react, they produce a large amount of carbon-dioxide. This gas is lighter than most other gases in our atmosphere and so they rise, filing the balloon.