Being an FTP server means that your computer would be listening (server=listen) for an FTP client wanting to connect. The server would ideally then ask for a username and password to allow the client to connect. The client could then browse anything in the FTP server's allowed download/upload area. That's the overview of what an FTP server does: it just listens for connections, authenticates them, and facilitates the actual transfers.
Now, the bad part. There is no security inherent in an FTP server and its transfers. Even the password is often transmitted insecurely. Now, your easiest way around that may be to make sure that you securely zip up any files you wish to transfer. Make sure to use a long, strong password.
Next, you'll need to have some way to know where your FTP server is, if you are hosting it yourself. Typical home Internet service changes your IP address (your unique identifier online) often. This is the address that FTP clients would need to know in order to find you. You can *buy* a static IP which does not ever change from your provided sometimes, but it would likely be cost-prohibitive. A better solution is to use a dynamic DNS service such as dyndns.com. This type of service allows you to set up a domain (sub-)name for free which is always up-to-date with whatever your current IP address is. So, you might be able to point FTP clients to novasolutionsftp.dyndns.org. You can pay for their service monthly at use your own domain name, btw.
So, the 'client uploads' area on your site would then point to your dynamic domain name, and you would put that domain name into any FTP client, including the one inside your transcription software.
So, FTP is quick and not really fraught with too much complexity, but it is not secure itself. There are such things as SFTP (Secure FTP) and FTPS (FTP-SSL), but both of those require a lot more configuration than we can cover in the quick reply box.
How's that?