Hi. There is no exit button, you just back out of the app sending out to the background. Doing this closes the app, howbeit of you are paying a song it will continue unless you pause or end the play queue.
This is a typical behaviour for Android as it will handle the memory and running processes by the system so you don't need to.
Please use the search feature before posting a problem/bug/idea. What's a Spotify Rock Star, and how do I become one? Using Spotify Since March 10th 2009 Twitter: @mechanimal82