This is an expression which shows your sympathy for people who are in pain or in trouble. It means "I feel sorry for them." For example, you can say this about people who have lost their homes in a disaster:

My heart goes out to all the folks in Missouri whose homes have been damaged in the flood.

This is an expression that people use in public situations like in front of a church group, on social networking sites, in emails, and so on. In face-to-face communication, you can say something like:

I feel so sorry for all those people!