Recover encrypted files without Migration Assistant

I have a hard drive from my old Mac that won't turn on. I'd like to recover the files from the hard drive, but they are encrypted by FileVault. I don't believe that the Migration Assistant is an option because I've had my new Mac for about a year, and would hate for the Migration Assistant to turn it into my computer from 3+ years ago.

I have a USB cord that connects and powers up my old hard drive (which I removed from my old Mac), so I can see the files in my new Mac, but can't seem to open them even after I enter my password (or what I believe it is). The files are "[username].sparsebundle". One of the Mac support representatives said that I can't open these files without migrating everything over to my new Mac. I thought that maybe I could migrate them to an external hard drive, or possibly partition my new Mac, and migrate the files to have a dual boot system. I don't know. I've never done this before. Any ideas.