By itself hydrogen will not explode. Even a mixture of hydrogen with air or pure oxygen gives no reaction. But when these mixtures are ignited a fast exothermic reaction takes place (see applications).
Hydrogen and oxygen are used to power the biggest space rockets because they provide the most energy per ton of any fuel. In order to get plenty on board, the gases have to be liquefied and kept very cold until needed. Rockets need to take their own oxygen for burning the hydrogen; when we have hydrogen-powered cars, we can use the oxygen in the air.
Things you can try yourself
You can do your own electrolysis of water using two bits of a wire and a battery. If the water has a little salt in it, so much the better, because pure water is rather a bad conductor. You should be able to see the bubbles of hydrogen and oxygen rising from the two wires where they dip into the water.