When I was in my early 20s I worked as pastoral support in boarding schools
Sometimes it's thinking that private education is better than state education and will set their child up for success in the future. It's an investment in their child's future
Some boarding schools open doors simply based on name, allow networking that can be beneficial in the future. Theres a reason why Eaton, and Harrow for example are over represented massively in things like lists of prime ministers and its not solely based on quality of teaching!
However there's an assumption that paying for it means that it's better quality. It's drummed up everywhere from celebs sending kids, to being woven into children's books as a sign of wealth, status and aspiration. All parents want their kids to have opportunities they didn't, and to have opportunities the parents didnt have as children.
One locally to me for example really ponders to the aspiration of being able to send your kids to a private boarding. Theres polo pony's on the lawn, lots of activities that you don't get in state ed. You can really see that parents will scrimp and save in order to send their child, because this looks like the boarding school they've read about in books, and the state school down the road is concrete and shabby
The actual school it's self is fairly terrible, massive drug problem, poor ratings, not particularly good grades coming out and lots of the teaching is done by unqualified staff in a way that isn't allowed in state schools. However parents get blinded by the dream of sending their kids to a school where the possibilities look endless