Complex sentence: During this cruising phase, an on-board system would collect air from the atmosphere, from which liquid oxygen would be separated and stored and used to burn the stored hydrogen in the final flight phase to attain orbit.
Split into simpler sentences: During this cruising phase, an on-board system would collect air from the atmosphere, from which liquid oxygen would be separated and stored. The liquid oxygen collected would then be used to burn the stored hydrogen in the final flight phase to attain orbit.