I grew up in a very WC family. My dad left school at 12 with no qualifications and my mum wasn’t allowed to sit the 11+ as she was Catholic in Northern Ireland (this was in the 50s) their advice was keep your head down and get a job asap, university isn’t for the likes of us.
So I did my a-levels and got a job and realised there was so much more available to me than a dead end office job. I was lucky as I worked for a big CP company in admin and blagged them to support me in doing some professional qualifications, then used that to go into a Big Four, where they supported me to do a MSC.
Whilst I was doing this I got the cultural education mentioned above (I’d never eaten out, been out of the UK, seen a play, been to a museum, used chopsticks - particularly embarrassing memory where I had to ask for a fork at a work lunch), I remember my first work trip abroad, I was so worried about the airport…
So fast forward to my children, we do lots of free museums in all sorts of topics, eat out as much as we can, travel when we can and talk about all the interesting jobs that I hear about. I also mentor girls via social mobility scheme at work, as I want to pass on what I have learned.
In short OP, it is a lot of things - education, aspiration (for your kids, and your kids need to have aspiration too) role models, mentoring and experiences. Finally it’s a willingness to now just fit in, that’s the one thing I am trying to drum into my girls, don’t be like everyone else, dance to your own beat.