Obviously this is a scam. I got the letter too.
A legitimate credit card offer, even if pre-approved, will still have an application with it. Or a link to the website to put in your information. In the rare case there is a fee, it would be on your first bill. No legitimate unsecured credit card ask for money up front.
Bottom line, if it seems something isn't right, it probably isn't. Fortunately these days we have the internet and sites like these to help avoid these scams.