By default, the Nikon D5300 displays your photo for 4 seconds immediately after it finishes recording the picture data to the memory card. This feature is called Image Review. To take a longer look, press the Playback button to set the camera to playback mode. Your photo then appears for 1 minute, after which the monitor goes to sleep if you don't press any buttons.
This auto shutdown is a Good Thing because the monitor is a major drain on the camera battery.
If necessary, you can adjust the playback and Image Review duration from the Custom Settings menu. (Note that whatever setting you use for playback shutoff also affects the auto shutdown of menu displays.) To access the relevant settings, select the Timers/AE Lock menu option and then choose Auto Off Timers.

You're offered these three general settings: Short, Normal, and Long. But these settings apply a set of timing options to the Live View display and exposure meter display as well as to the playback/menus and Image Review displays. It’s recommended that you select Custom and then press the Multi Selector right to display the screen shown on the right. You can set the Playback/Menu timing and Image Review timing separately.
To turn off Image Review altogether, head for the Playback menu and set that option to Off.
