"pining" = to miss somebody to yearn/long for them to come back.
"pining" refers to a mental state.
"pinIng her heart way" is a term we use to describe someone who is in emotional turmoil over the loss of a lover, partner, friend, family member, pet etc.
The loss can be because of death or a break up.
It describes the initial period and behaviour of people suffering such loss. Commonly they lock themselves away from others and do not leave the house for a while.