Make sure you have a stable internet connection running (ideally WiFi), then terminate the Music Maker JAM app completely (by swiping it off the list of running apps) and restart it again. It should pick up on the Internet connection and automatically proceed with the download of the style.
If you're intending to use a data plan/connection to perform the content download, please also make sure that the setting to restrict downloads to WLAN/WiFi hasn't been enabled for Music Maker JAM. You can find this under the app settings for Music Maker Jam in the system settings. Turn it off to enable the app to generally download, even outside of WLAN/WiFi use.
The styles will only continue to download while the MMJ app is actively open (not just running in the background) so to complete the downloads you will need to stay in the app.