The ending of Batman Forever settles three things at once: Two-Face dies, the Riddler loses his mind, and Bruce Wayne chooses to keep fighting crime with Dick Grayson as his partner. Two-Face falls to his death in the final fight, and the Riddler's machine fails, leaving the Riddler locked in Arkham Asylum, where he insists he is Batman.
Why does the Riddler say that? The film has already established that his device read Batman's identity out of Bruce Wayne's mind, so the claim is the broken remains of real knowledge. In my opinion, the joke is that nobody in the asylum can use it, and the villain who beat Batman's secrecy ends up as the one person who cannot make anyone listen. It could be argued that this is the film's neatest idea, because it turns a mind reading gadget into a punchline about madness.
How does Two-Face actually go? He falls after a coin trick backfires, which is a fitting end for a man defined by chance, and it is the film's one clean piece of villain logic. Compare that with the Riddler, whose own invention overloads and unravels him. Both villains are undone by the tools they built their identities around, and that symmetry is Schumacher's main structural idea. It is also a different shape from Batman Returns, where Burton gave Penguin and Catwoman a tangled, shared downfall tied to Max Shreck's schemes rather than to a single gadget.
The human ending is Bruce and Dick. Bruce struggles with his past throughout the film, and Chase Meridian presses him about his double life, so taking Dick on as Robin is his answer. Dick arrives wanting revenge after Two-Face kills his family at the circus, and Bruce takes him in, which puts the new partnership on a very different footing from simple sidekick work. Perhaps the point is that he chooses a partner over carrying on alone, and that he sees his own grief reflected in the boy. Keep in mind that Burton stayed on as a producer while Schumacher directed, so the film carries the earlier series' Gotham, Alfred and Commissioner Gordon into a brighter, busier story.
That sets up Batman and Robin as the next step in the series, where the partnership is strained and the same director returns. If you are watching in release order, this ending is the natural hinge between the Burton films and the Schumacher pair.