It's a little more complicated than just opening up a bunch of ports. Not knowing your network/router setup, let's assume your computer is at 192.168.1.100 and you router is 192.168.1.1. Let's also say your router's public IP from the ISP is 123.4.5.6.
Using PASV mode, your computer opens a connection to the FTP server using port 21. It issues the PASV command and the server responds using the open connection on port 21 with the new port number that it will be listening on. Your application, then, will try to open a connection to the FTP server using the port that the FTP server will be listening on. This port will be where the transfer will occur.
Since your host is the server, here is what happens using the addresses we assumed above. Remember that your router is doing NAT and anyone outside of your LAN only knows you by the router's public IP address of 123.4.5.6.
[ol][li]The Internet Backup app starts (?) and opens an FTP connection with your host. Not sure how this connection starts and who initiates it. [/li]
[li]Since it is PASV mode, your host responds that it will be listening at 192.168.1.100 on port 12345, for example.[/li]
[li]The backup app tries to open a connection to 192.168.1.100 and port 12345 but can not do it since your router is translating the address[/li][/ol]
There is likely more involved since I don't know the details of your environment but I would guess this is pretty close.
Hope that helps.
The Old Man