Not quite ... explosives do not need external oxygen to work. An oxygen supplier and the stuff to be burned are packed together in an inert way (e.g. gunpowder: the nitrate supplies the oxygen, the coal and the sulfur get burned).

As for the extinguish-fire-with-explosives trick: every fire needs three conditions: burnable material (here: oil), oxygen (here: surrounding air) and heat (here: the flame). If you take one of these three, the fire will cease. The explosives simply blow away the flame and any heated material around it.

Sorry if this sounds like a lecture, but fire's been a special field of interest of mine. ;-)

Andreas