whether a father's interest in his sons has much to do with how well his sons turn out
Has much to do with: is related to; has a correlation with; corresponds to.
How well something turns out: What is the quality of something, once it is finished being produced. (In this case, do the sons grow up to be good, honest, contributing members of society.)
The overall sentence fragment is presumably either asking if, or stating that, the amount of interest a father shows in his sons has a clear effect on how well the sons do once they have grown up and moved out.