Its not entirely clear why Diana chooses to stay in the outside world at the end, but I think there are two theories that seem to have the most support from the movie:
She Has No Choice
When she leaves the island with Trevor, her mother tells her
If you choose to leave, you may never return.
Now, there's two ways to interpret this. It could be simply her mother expressing worry that Diana may get killed, and never come back home. But given what we know (and Hippolyta knows) about Diana, it seems odd for her to be worried about that.
On the other hand, she may be telling Diana that if she chooses to leave the island, she can never return. There is absolutely no explanation why that would be true, unfortunately. Clearly, the island can be reached by normal means, and the Amazons are worried about more Germans arriving. Also, Hippolyta doesn't use this same objection to sending other Amazons out to fight, she only brings it up with Diana. But we know that Diana's situation is special, so it's not unreasonable to think special rules apply to her. For one thing, she's not really an Amazon, so perhaps there is something tying her to the island that will break when she leaves. Or, perhaps Hippolyta means that Diana would draw enemies (like Ares) back to the island with her, so she would have to stay away for everyone's good.
She Has A Mission
To me, the more likely explanation (and the one I think fits thematically with the movie) is that Diana found her true purpose, and it meant staying in the world.
During the bulk of the movie, Diana believed that her mission was to kill Ares and end war forever. She believed that this would return human kind to the pristine kind and peaceful nature it had before Ares corrupted them. Of course, that turned out not to be true, but by the end of the movie, Diana had found something else to believe in besides destiny.
She had seen how horrible people really were, Ares notwithstanding. But she had also seen how good they could be, if given the chance. And she had seen their capacity to love each other, and how powerful that is. She had decided that, despite her mother's protests, humans maybe did deserve her help after all.
So, she chose to stay in the outside world because she believes that to be her true purpose. She is one of the people who can genuinely make a difference in the world, and help make it better, and that's what she chooses to do.
On a side note: there was almost 100 years between Diana's appearances in the DCEU so far; we don't know that she never went back home. Perhaps we'll find out in a future movie that she does visit, but chooses to live in the outside world for personal reasons.