Legit contests which have entry fees have those fees because they get too many entries. Sure, they may offset some expenses, but basically the payment exists to make people think twice about entering. It's a basic "must be at least this serious to enter" bar.

The scam, on the other hand, is the opposite — they've reached out to you, and expend a lot of effort getting you hooked and then ask for money. Nothing legitimate ever works this way. Your instincts are right on the nose. Run away.

Like many scams, the basic hook here is an appeal to your ego — or vanity. It starts with someone complimenting your artwork, or your writing, or your business skill, or whatever. Human nature being what it is, we want to believe — which makes it easy to set aside skepticism and ignore warning flags.

On this particular scam, check out this forum thread for some people with personal experience, as well as some forensic investigation. Some choice quotes from that thread:

100% a scam. I reached out to artists who "won" (basically anybody who pays). People are told different things (pay to be promoted, art contest with distinguished panel of judges, magazine sold in Amsterdam, London and New York). Some bought copies of a magazine that never arrived. ISBN numbers of magazines "published" belong to other real publications.

and

I can now tell with 95% probability that 1340 art is a scam - don't pay money for attending into their odd competitions.

Currently at address of 1340 art in Netherlands is actually located Amsterdam Art, which has nothing to do with 1340 art.

Further, I note that the site touts a 5-star "Trustpilot" rating. Ironically, when you see this kind of thing front and center, that's a warning sign. Trustpilot is a real, independent thing, but they're famous for being full of false reviews. Indeed, the link on the site itself (which claims 5 stars and 8.8/10 as a rating) isn't clickable, but if you go and search, you find a three star 6.8/10 rating — and a long list of short, suspicious five-star reviews punctuated by one-star reviews from artists who were taken in. Take a look at the overview:

This is not a ... normal company profile.