Any time you stream a new meditation or story, the audio file automatically downloads to your device in the background. This ensures that the file will be cached for future listening, so that you can enjoy the sessions offline. When you listen to Day 1 of a multi-day program, the remaining days' sessions will be downloaded right away as well. (The exception to this is the Daily Calm - as these sessions update each day, they are only available for streaming over an internet connection.)
You can also download meditations and sleep stories by tapping the 'Download' cloud icon in the top corner of the program screen.