Boost Rush mode is not available until after you complete the story for the first time. The mode has no enemies and is essentially steering the ship through a cave to avoid obstacles while in a permanent boost run.
At first this seems challenging as one mess up forces a restart and you run out of space pretty fast trying to squeeze past large rocks. After multiple restarts I realized that you CAN "push" the ship into the ceiling and floor without crashing, only head on collisions cause death. Using this method to navigate the tiny openings I finished the mode in one try.