Transitioning between two scenes can take many forms. For a simple game, you could just throw up a results message, then unload that scene, and load a new one. The transition would be virtually instantaneous and not disruptive, though you could insert a small load screen, just to produce a brief break between levels. This would also allow you to add some more graphics in order to add to the general atmosphere of the game, adding commentary, jokes, as well as many other options.
A transition can be started by reaching a certain part of the scene, entering a portal, or accomplishing a certain goal in the scene, such as collecting a certain number of points.
Upon reaching this trigger for a scene transition, we can show a message regarding the player's success in the level and then shift to the loading screen or transitioning effect, depending on our stylistic choices and how quickly we can load the new scene's resources. Preloading the new scene's resources upon entering the results...