Or is it just a basic familiarity (like a couple of most common words) and awareness that English exists?
EDIT: I see from a reply below by Freak_NL that it probably means before the kids start learning English at school. That makes more sense, as they would be older at that point.
I don't know about The Netherlands but here in Norway children start learning English as soon as they start school at the age of five or six. But quite likely many of them will have at least some English already because of English language television, computer games, etc.
It took me several years of school before being able to read fast enough to follow along subtitles, and the same for everyone I know.
This used to be the case in the Netherlands too; I picked up a significant body of English from British TV series watched with subtitles as kid. Nowadays this advantage will probably be missed by most children, because the streaming services offer a lot of dubbed content, and you get to pick what you watch unless someone guides you. Subtitles can be avoided for longer.
Since kids first learn their native language ( write, read and speak) in school and only years after then ( mostly), learn foreign languages.
When they learned to do it in their native language, they hear English spoken on tv with eg. Dutch subtitles and pick it up. Sometimes before they have English lessons.
Most kids, as such, know a fair amount of English before they have it ( = English) in school.
The Dutch subtitles isn't always a requirement though. Kids will pick it up in some shows, eg. Pokemon would be a good example if English spoken.
Why wouldn't swedes?