No June made a deal with her brothers or something. She was supposed to bring the person she loved and who loves her in return back to the sea where they live. That's why when she "failed" to find true love (because yeon jeong got married out of fear what others thought of her), the brothers offered her a sword to stab yeon jeong and drink her blood. But in the end when June jumped back into the ocean, yeon jeong jumped after her proving that June did find her true love. So yeon jeong probably turned into a mermaid by using their magic or something. Remember when June promised yeon jeong if she followed her, they'd live everyday happy? I think it's an open ending for the reader to decide if she died by drowning or if she went with June and became a mermaid but that's my take on it.
So did she end up with June at the end in the ocean. Or did they both die?