Why, or rather, HOW is this the case?

Because you need something to burn, i.e. something that can be oxidized.

The fuel burns by combining with the oxygen. Remove either the fuel or the oxygen and the fire is extinguished.

 

(Wasn't there an almost identical question recently?)

'Non-flammable' is a pretty loose word. It's meaning can vary a bit with context. Many materials that are described as non-flammable can become decidedly flammable in different atmospheres and/or at different temperatures and pressures.

 

If they're hot enough, exothermic chemical reactions in the gas phase can emit some of the energy produced in the visible spectrum. We see this as a 'flame'.

 

Due to the make up of our atmosphere, one of the reactants is usually oxygen, but it doesn't need to be. The reaction between hydrogen and chlorine (among many other possibilities) gives a pretty neat flame. It's just that we don't see these other reactions so often.

It takes two to tango.

In an atmosphere of hydrogen, oxygen is flammable.