The bandwidth saver exits a streaming app after 4 hours with no input from the user and returns the Roku to its home screen. This it to prevent the stream from continuing to download unwatched and waste bandwidth. At that point, if there is a screen saver configured on that Roku, the screen saver will kick in at its prescribed delay time, otherwise it will stay on the home screen. Bandwidth saver does not put the Roku "to sleep".
Roku operating system version 10.5 added added a new "Power" item in the "Settings > System" menu for some of the newer Roku devices. The new "Auto power savings" item there will put your Roku device into a low power standby condition after a period of inactivity, such as when your tv has been turned off for a while.
Pressing a Roku control key on your Roku remote (not Power, Volume, or Mute which act on the TV, not the Roku) should bring the Roku out of standby and back to full operation. Pulling the power for a few seconds to restart the Roku will restore full operation as well.
If you have a "Settings > System > Power" menu item, deactivating the power saving option you find under "Auto power savings" will prevent your Roku from going into standby in the first place.
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