[NickPollard]

I don't know the maths involved, but the idea presumably is that they save more weight not carrying (as much) oxygen than they lose by adding the rest of the engine. By being able to draw oxygen from the air whilst in the atmosphere, they only need to store a smaller amount of oxygen for when they hit space. This is the main saving I believe.

[pja]

Indeed. O2 is much heavier than H2 so if you can scavenge the O2 out of the atmosphere on the way up then the amount you need to carry for the final burn in space is much smaller & the weight savings are huge.

Consider that the Space Shuttle external fuel tank contained 650,000kg of O2 and only 100,000 kg of H2. Save 10% of your O2 needs and suddenly you've got 60,000 kg to play with. The Shuttle payload was only about 25,000 kg.