The final is an alternate ending/fragment, where Takano's parents don't die as a result of the "miracle" born from the combinated wills of Rika and her friends. In the original ending, the miracle occurs instead when Takano shoots at Hanyuu in the mountain forest and misses her shot.
The adult version of Rika is most likely Frederica Bernkastel. Her real nature/identity is often questioned and discussed by fans, most notably those who know the series Umineko no Naku Koro ni by the same author; however, a direct and clear link between the Bernkastel from both series is not established.
From a simple Higurashi point of view, at least, Bernkastel is an entity born from Rika's "mature personality" due to her many reincarnations. It is speculated that while Rika is still in the fragment where she won with her friends and live a peaceful life there, Bernkastel was able to become a separate existence that can freely travel between the different fragments (= alternative realities).