Douglas recounts his childhood experience at the Y.M.C.A. pool to enable the readers to understand the exact nature and intensity of the terror. The fear of being surrounded by the water, the fear of putting his head in the water, the fear of choking and the fear of his limbs going numb couldn't have been explained to a reader unacquainted with Douglas' childhood experience. In that case, the elaborate strategy adopted by the author (and his instructor) and the time-taken by him to learn or master even simple things, though put in the perspective of his fear of water, couldn't have been understood properly.
By quoting Roosevelt, "All we have to fear is fear itself," Douglas indicates the larger meaning that he draws from his experience. For him, the importance of life became evident when he encountered death or rather its proximity threatening his life.