It is to express sympathy or condolences. It could be said in the event of a significant illness/injury, accident, natural disaster, or death. It could be expressed to the person who is suffering, the family members of a deceased person, or generally anyone affected by the problem.

The dictionary is correct. The expression has a range of meanings, but they all have to do with sympathy and empathy. It is not used to indicate romantic attraction or romantic love. Two dictionary definitions of "heart" include: "The vital center and source of one's being, emotions, and sensibilities." "The repository of one's deepest and sincerest feelings and beliefs: an appeal from the heart; a subject dear to her heart." There are many expressions with the word "heart," and many words related to it. Most of them carry ideas like "deeply felt," "sincere," and "intense," or "non-romantic affection." Most do not carry the idea of romantic love.

One possible interpretation is indeed sympathy, but there are other possibilities. It really means that you stand with them in your heart, the figurative seat of your emotions. It might not be sympathy at all. Here's an example. Suppose your child has a very good school teacher who seems able to work miracles with her students. You are so happy that your child has that teacher. You might say "my heart goes out to her for all the hard work she does to help the children."