Then he found out she got engaged, and he made the rest of the song super difficult so she could never play it.
You can listen to the song and hear when his heart breaks: https://youtu.be/CQIDLhBJ2y4
It is very unlikely that this story is anything more than a fanciful tale: The dedicatee of the piece is unknown; the work was found in papers long after Beethoven's death and was unknown before that. So, the prospect of there being someone who was told the story and wrote it down, even though it's recorded nowhere is close to nil. See the Wikipedia article for more details.
Another fun theory I’ve read is that the piece was called “Für X” for different values of X at different times.
According to this theory, Beethoven intentionally didn’t publish it so that he could continue to use the same piece to impress different women, “Look what I wrote for you”.
Efficient.
Bo Burnham, Repeat Stuff: https://m.youtube.com/watch?v=nt9c0UeYhFc