Private schools often have selective entrance requirements, so they're mostly taking pupils who are likely to do well to begin with.
Pupils who go to private schools tend to come from families of rich, educated people and high achievers, and are more likely to see this as the norm. They're less likely to be satisfied with average or mediocre achievements, and are more likely to consider education to be a priority. (e.g. a pupil from a family of Oxbridge graduates might consider it of paramount importance that he gets into Oxbridge himself, so that he lives up to the norms and expectations that are naturally upon him - whereas a pupil from a family of people who left school at 16 might be overjoyed just to get into any university, because even that by itself will do his family proud.) Their parents are also likely to give them more encouragement and put more pressure on them with regards to their studies, and bring them up in such a way that education is their main priority, and their life revolves around it - because that is how they are likely to have been brought up themselves.
Similar to the previous point - even if a person does not come from a family of high achievers, if he goes to a private school (e.g. on a bursary or scholarship), he will be surrounded by other people who are from families of high achievers, or who are high achievers themselves as a result. They're then less likely to be satisfied with average grades, when everyone else around them is getting brilliant grades. (You can see this in the way that many state schools propagate the idea that a "good GCSE" is any grade from A* to C, and that an acceptably successful student is one who gets at least five of these. Whereas in most private schools, a person who only got 5 C grades at GCSE would probably have the worst grades in his year, and would probably also be asked to leave.) How hard you work will depend on how important you think it is to do well, and also how high you set the bar. And a person whose friends are all getting 12A*s won't set the bar at 5 C's.
It's possible that to some extent, the intelligence required to do well at school is simply inherited. People who can afford to send their children to private school are likely to have highly paid jobs, which require intelligent people to do them. And so their children might just turn out to be naturally more intelligent. Although this is probably a weaker point, as intelligence alone is certainly not enough to succeed with GCSEs and A-Levels.
Private schools have the money to pay for good teachers, better learning equipment and resources etc. and the parents of private school pupils are more likely to be able to pay for extra tuition, books and learning resources at home, and may be able to help educate the children themselves. Again, I think this is a weaker point though. Money and resources alone aren't enough to succeed with GCSE's and A-Levels either - your teachers and parents can't sit the exams for you.
I think the main reason is the one discussed in my two largest paragraphs - that pupils are more likely to work hard if they have high aspirations. Everyone wants to be successful, but different people have different definitions of what "success" is. But it will usually depend on who you're surrounded by at school, and at home. If you're always surrounded by highly educated people, then "success" will probably mean getting a good education, and you'll want to work harder towards it.