Like the iOS App Store, the Mac App Store allows you to re-download materials you've already purchased without having to pay again so long as you use the same account ID. You'll probably want to get a list of all the applications you've purchased, so you can restore them. To do so, choose Store > Purchases from within the App Store application. From there, you can click the Install buttons listed next to each app purchase. (Thanks, Kyle)
Unfortunately, re-downloading software from Mac App Store outside of the Store > Purchases screen acts as if you're buying the app from scratch. Apple doesn't quite get to the point of checking that you already own the app until after you have agreed to let your credit card be charged for the "purchase." Only then, do you get the comfort of this free update alert.
This is poor user experience on the part of Apple. It really should check for prior purchase before making you agree to have your card charged. Hopefully this will be updated in new releases of the App Store and in iTunes, because both stores really do make you think that you're going to have to pay twice, when you don't need to.
That said, you then have to get the application downloaded, and that can also be problematic because when purchasing an item a second time, experience has shown that the new download sometimes encounters issues, e.g.,
If you encounter the dreaded 403 Forbidden error, you may have to contact Mac App Store/iTunes customer support. The account and billing support line requires that you leave a help message at the site, but they do get back to you quickly (usually within 24 hours), and the staff is probably the most supportive and helpful part of the entire Apple organization.
So, that is everything you probably need to know on the re-download front. Good luck, and here's hoping everything proceeds smoothly for you.