You need an oxidizer in a rocket, because rockets go to space, and space is a near vacuum, meaning it is emptiness, resulting in pretty much no oxygen being present. To create combustion/a fire, you need oxygen. We have plenty of that on Earth, which is why & how fires exist, but in space, you need to bring your own with you, so rockets carry Liquid Oxygen. Liquid because it is more dense than gaseous oxygen, so you can fit more of the oxygen in your fuel tank. When Hydrogen and Oxygen burn together, the exhaust is simply just water vapor, and nothing else. That's right, the exhaust is gaseous H2O. If you capture the exhaust, and liquify it (simply by letting it sit there to turn into a liquid since water is a liquid at room temperature), you can literally drink it.
The image at the beginning of this post is Hydrogen & Oxygen, so none of that is actually pollution, it basically just makes clouds, and if you look in the middle of that big white cloud of steam, and then look to the right, you can see a grey area, and that is water raining down from the cloud generated by the steam back down onto the surface of Earth.
Although this sounds great, and that we no "no longer need to pollute" to get to space, and can simply use Hydrogen, well, yeah it is a great idea. But that is to just get to space. Hydrogen/Oxygen is definitely not the best fuel choice to get to Earth orbit or beyond, some may say it isn't even good. When you burn rocket fuel & oxidizer, you are usually burning hydrocarbons. This is the most powerful type of liquid (or gaseous) fuel. A hydrocarbon is a molecule which contains hydrogen atom(s) and carbon atom(s). Hydrogen is an exception as it is just H2, 2 hydrogen atoms, and no carbon atoms, the simplest fuel, but not the most powerful. Hydrocarbon fuels are more powerful than simple Hydrogen fuel.
Coming right after Hydrogen, in 2nd place, CH4. CH4, aka Methane is a hydrocarbon consisting of 4 Hydrogen atoms and 1 Carbon atom.