Simply put, Fedora will not automatically recognize this adapter. USB wireless is still not perfected in Linux. In order to use it, you would need to install one of tywo packages that will allow Fedora to see the adapter, either the Linxant package (which costs about $19.99 - it is a commercial product) or ndiswrapper. Then you can use the Windows drivers from your cd drivers disk to get the card recognized. These two linux packages are designed to work with the Windows XP drivers that came with the Linksys disk.
Just as in Windows, you need to install drivers for the adapter to work. You also need to know what the settings are for your router... the essid name, and if you are using encription, the WPA or WEP key for the router or modem that you have.
I actually have one of these cards (version 4) which is a little different, but I may give it a go this week.
If you are interested, I will report back my (hopefully) success.