Drupal 7's default behavior is to redirect the user to the full display of a node after it has been saved.
If you want to stay in the edit form after saving, add this snippet to a custom module adjusting the module namespace in the two functions.
For Drupal 8 or 9, have a look at the comment below by @zanvidmar.
Thank you for the code snippet. I am not sure for which core version this applies. However since $form_state is an object, this code snippet should be updated to: