In that department means with regard to that or in that area. For example: I'm good at languages but not at math - I need some help in that department. In that post, the department being referred to is helping the children with their homework. Dad has learned his lesson in that department so he has learned not to help the children with their homework. "Bailed out" means he helped him out with a difficult situation, so his dad helped him out a million times before with his homework. "jump right in" means to do something instantly "department" here means area and is referring to how familiar he is with this. "learnt his lesson" means as the dad has helped so many times before, he has decided not to help his son this time. So, because his dad has helped him out lots of times before, this time his dad decided not to and instead it it up to his son to sort this problem himself.