Translation:Their expenses for their child are large.
"Their expenses for their child are big" sounds like bad English to me. It sounds a little better with "large". I'm not saying the English sentence is wrong, but why does it sound so weird??
They are both fine, but you are correct. Big refers to the size of an object. Large refers to size or amount; hence,expenses are large.
I think a lot would sound better, but I've heard locals say big money.