It is true that hydrogen is an explosive gas, and it also needs oxygen to burn. But the key point lies in this question: What is burning? The answer is that burning is the process of reacting with oxygen to produce energy. (Well, to be really pedantic, it may be possible to use other gases such a Fluorine, but oxygen is so readily available, we usually mean oxygen.)
The burning process in the case of oxygen and hydrogen is relatively simple. If you put together suitable volumes of hydrogen and oxygen and provide a spark to start the reaction, one oxygen atom will combine with two hydrogen atoms, and will release energy in the process. The energy gets released in the form of molecular kinetic energy, and since the motion is random, this is exactly what we call heat energy. The gases heat up, and as all expanding gases do, they expand. That fast expansion of hot gases is what we call "an explosion".
What is the result of this burning/explosion? The answer is simple, if the proportions were right (one volume of oxygen for two volumes of hydrogen gas) all you get is water! Thus, water is _already burnt_. It is the "ashes" of hydrogen after it has burned.
If you wish to burn it again, you have to separate oxygen and hydrogen. But, to do that, you need to supply as much energy as is released when the burning occurred. This is possible by electrolysis.