That means he is crying a lot (crying uncontrollably)... It's a very common expression and is informal language. Hope that helps!

He's crying uncontrollably, like crying a lot, "sobbing" is also a term that refers to someone who's crying loudly

This is a very emotional moment for the man. "Sobbing his heart out" means that he is not only crying uncontrollably but that very deep emotion is felt. He is very sad, but more than this. He is hurt so badly that it feels unrepairable at this moment.

It means that the big strong man was crying uncontrollably.

Sobbing can mean crying...so basically "crying his heart out" crying deeply...expressing deep hurt or pain he felt...maybe his heart was broken?