It's the same in English (and Swedish). I guess it's just the best way to describe words that aren't easy to say or feelings difficult to convey. They just feel heavy to bear and hard to hear, so I'm assuming it has the same connotation in Japanese as in the aforementioned languages (although I cannot guarantee that I'm correct). In English when someone has said something that is difficult to say they may often describe it as "a weight being lifted off their shoulders" afterwards. It's the same as describing a conversation as being "deep" or "light".

That's kind of metaphor. Heavy words or sentences restrict acts, thought and something which make us feel pressure. I often hear these from couple like this: "Don't talk other girls!"," I can die for you."