That English translation seems a bit too literal to me. The German expression "jemandem fällt ein Stein vom Herzen" means that somebody is relieved of a worry that had bothered them greatly. An equivalent English phrase would be something like "That took a load / a weight of somebody's mind".
The German figure of speech is stronger figurative than the English one. Imagine a heavy stone that's weighing on your heart, on your feelings, that keeps you uneasy. And then that heavy stone is gone, it fell down from your heart.
As a side note, there's another phrase where German uses the heart, while English uses the mind. If you can tell that somebody has something on their mind, you might ask them "Was hast Du auf dem Herzen?".
Thank you both - Henning Kockerbeck for the explanation as I wondered if the German sentence was actually correct, and indeed it makes sense!
Lisa-Lingvist, like I said I was on my phone and explanations like that never seem to show up, so I thought there was a chance I was missing something. It's just that this time my doubt was too great to wait until I did it again on the computer. (Also some sentences are truncated on the smartphone, FYI it's an Android phone).
Thanks for the swift replies!!