I think the posters saying don't worry, you're just moving to another house, it's not necessarily better/better school, no need to explain are short-sighted.
I've been in this exact position op. When dh and I first bought a house we looked at the house and how close it was to work - school catchments didn't figure. Then we had dc. Our local Primary, whilst in a very deprived area, was lovely. However, we realised by about Y3 that the catchment comp was anything but. Awful GCSE results, run down, constant trouble.
We moved when dc were in Y6 and Y4 (just in time for school applications). About 10 miles within the same County - but from an area classed as significantly deprived to the 'posh area' with the best performing school, which is well known as such.
There was no hiding this fact from neighbours or school parents. No casual 'oh we just like the house' possible, and it wouldn't have been believed. House prices in the new area are 3 or 4 times the price of old area and people know this. Many pupils from the new school end up at Oxbridge and people know this. And there was a LOT of inverse snobbery. People from our area just didn't entertain moving to X area - it was out of reach for everyone we knew.
We had lots of questions and curiosity and it was uncomfortable. It's awkward, theres no denying it. We went with along the lines of 'Oh we've always loved X area, always wanted to live near the sea. It's taken about twenty years to afford it though! The house prices make me want to cry so we'll be on beans on toast for the next ten years haha'.
It seemed to land OK. Theres just no point in trying to ignore what many people are really asking which is 'how the fuck are you affording that, you snobby twat?'
For the dc, we were honest. We explained that we wanted to give them the best opportunity at school and life and that X school was the best one and in a far nicer area, so we were moving. But we also explained that some people we knew would find that very hurtful and that it was best not to say that to others, and they were mature enough to understand this. They stuck to 'parents are making us'.
I would really, really encourage a rethink on not moving Primary though.
Dc1 started in new Comp straight from Y6, the only one from his school. He massively struggled for two years. Seeing this, we moved dc2 at the start of Y6. He came around quickly, had a year in Primary in our new area and did the transition to Y7 seamlessly. We absolutely kicked ourselves that we didn't move Primary too.