Hydrogen has a great affinity for oxygen. They combine together to form water H₂O, and in the process a large amount of energy is released. There are two interesting aspects in the experiment of making explode the hydrogen balloon that I would like to call your attention to.

First, that pure hydrogen will not explode when ignited, it just burns with a quiet flame, no explosion. The explosion occurs when hydrogen has been previously mixed with oxygen (for example, because oxygen can diffuse from outside into the balloon). There is a range of mixture that will explode (something like between 6 and 60 % of hydrogen, but these numbers I am quoting from memory, if you need the exact range I would have to look them up).

Second: You can mix hydrogen and oxygen and they will stay together, apparently without reacting forever. Without some initial push (given by the flame of a match, for example), their reaction is too slow to be perceived. But if we give enough energy to get started, it will become an extremely fast reaction: an explosion.

Hope this satisfy your question, if you still have doubts, let me know.

I think that you're probably thinking of a helium balloon, since a hydrogen balloon will explode so violently to be dangerous to the person holding it. Any kind of balloon is filled up by gas (which could be ordinary air), which is under pressure. Heating the gas raises the pressure, as well as burns a hole in the rubber of the balloon. The rubber fails, and the balloon pops.

Now, if the balloon were only helium, then the gas would just get released. Hydrogen, however, ignites with the oxygen in the air to form water vapor, and burns extremely vigorously. The airship "Hindenburg" was an example of a hydrogen-filled balloon that got ignited and caught fire.