When baking soda and vinegar combine in a container with a balloon placed over the opening, one of the resulting materials is carbon dioxide, which fills up the balloon because it has nowhere else to go. Water and sodium acetate are also created from this combination. The mouth of the balloon needs to have a good seal in order to prevent the carbon dioxide from escaping.
When the mixture is observed, bubbles are visible as the two materials combine and begin to react with one another. These bubbles are carbon dioxide escaping from the mixture. Carbon dioxide is heavier than air, so it is important to use a large enough mixture that the carbon dioxide fills up the container and makes its way into the balloon. If there is not enough mixture, then the balloon does not inflate or only partially inflates. Alternatively, if there is too much mixture, then it is possible for the pressure to be too great. This can lead to the balloon being forced off of the mixture container or popping as it expands.
This is the same principle that creates the eruption in a vinegar and baking soda volcano. The difference between these two experiments is that in the volcano experiment the focus is on the bubbles themselves rather than the resulting extra volume of carbon dioxide.