To enable the sleep timer while viewing the Now Playing screen, tap the double line icon next to the Back 30s button, then tap on the clock icon (with Z’s). A new menu will appear from which you can select how long you would like Stitcher to continue playing until it turns off. If Stitcher is playing at the completion of the timer, it will go silent and stop playing.
The current version of Stitcher on iOS also includes a feature that disables continuous playback. To manage this, go to the app’s Settings > General > Playback Behavior.