It is very open in the sense that we share it with everyone we interact with. Thus, if we give our heart away, it could never be our whole heart, but instead only a part of it, because our heart is always changing with every person we encounter.
Giving your heart away means exposing yourself to another. When you do so, you create a bond with the person. This can never be broken because [once you give your heart away] you can never take it back.
When you give your heart away, you give it along with every piece of you. Giving your heart away is completely trusting another person not to hurt you.