They are made using Adobe Flash?
Uhh, there's two main types, the Wix ones and ones that are made by hand.
The Wix ones are done using the Wix generator, then put on the profile using a media section.
If that's what you've been seeing, there's a guide for them here.
The ones done by hand are created using some sort of SWF maker, often Adobe Flash CS-whatever (CS6 is current).
Adobe's can be pretty expensive, but if you're a student your school may have a copy you can use (even if it's just on the school computers), or you can get it at a discount. [Adobe Flash] / [Trial Version]
There may be other SWF editors out there, but I don't really know about them. Wikipedia has this to say on the matter.
Once you have the program, making the profile is a bit like using Photoshop - as in, the program may be tricky to get the hang of at first and relies somewhat on your artistic talent. (which is why I suck at it *shot*) There's also stuff like Actionscript involved to make Flash do certain things for you. The internet probably has a lot of Flash tutorials, some better than others.
Once your SWF is created, you have to host it somewhere, get its direct link, and put it in a media section on your profile. Then add in a bit of CSS to change its size and make it blend with the rest of the profile.
My favorite Flash profiles have been those that effectively combine Flash and CSS - something you don't see much of with Wix.
Dont forget, if you made one yourself. You also need a place to upload. DeviantArt allows one to put flash file, though I am not so sure about it. Getting free host-server like www.webs.com , fileden, or ucoz is a good idea for starter.