Not all independent schools are selective. Many (including the ones featured in the article) are highly selective, but many are not selective but still get results that are much higher than the local comprehensives.
I am about to start working in an independent school, which is non selective (and down the road from two highly selective independent schools who cream off the most able students) but still gets well above average results at GCSE.
I have a few ideas as to why this might be.
The classes are very small. The biggest class I have is 23 pupils (top set), and the smallest is 8 (bottom set), therefore the pupils get more time with the teacher to focus on areas where they are struggling, especially the less able pupils.
The discipline is very strong, perhaps even draconian. If pupils do not meet the expectations of the school, they are asked to leave. On average 2 pupils are expelled each year (out of 700). It is near impossible to expel a pupil from a state school. This means that disruptive pupils will not be distracting well behaved hard working pupils, who will be able to focus on their work more easily.
The day is much longer in many private schools. We finish school at 5:30 most days. With a longer day there is more time for teaching so therefore pupils will understand their course better.
Expectations are extremely high, of everyone. Teachers expect the highest standards of behaviour and effort, and pupils and parents expect the highest standards of teaching from the staff. It is normal for teaching staff to spend break times, lunch times, and even weekends (boarding school) supporting pupils who are struggling, especially just before exams. This would be unheard of in most state schools.