A feeling of an inability to understand, accept or fix something which causes one pain on a personal level. Most will equate a broken heart with the loss of love between people. This is but one aspect of experiencing a broken heart. Though, of course, it is not an actual physical thing; and this condition is subject to the individual experience; it is, none the less, a real experience. Feelings of loss, hurt, pain, disappointment and helplessness contribute to the "broken heart" syndrome, in my humble opinion.
When one feels overwhelmed and helpless in the face of something 'bigger' than themselves, when one cannort figure out a way to improve conditions; one is left with a feeling of being "broken," hurting, and impotent. Therefore, a "broken heart" is the result.
At least, for me, this is how I interpret the term.