Do you have more context? Without that, I would guess -the hearer wasn't really paying attention (e.g., "I was playing video games and only half heard my mother's list of items to pick up at the store. I think she may have wanted lime or maybe thyme.") or -the hearer couldn't hear clearly. (e.g., they're in a really loud place and it's difficult to hear the speaker) Either way, the hearer isn't really clear about what was said.

In addition to the other answers, it also implies that you were not paying attention or was distracted.

Well 'half' = 50%, If someone is telling me a story but didn't get to finish it, I only heard half of the story. It doesn't mean I only heard 50%, I just didn't get to the end of the story. Hope that helps.

It means that you would have not heard something properly.