gpsdrive can connect to your gps with two methods:
First, it can use gpsd to connect.
gpsd is a program listening to your real GPS device and providing the data to applications using linux sockets or tcp ports.
This method has the advantage that you can use multiple applications with your gps at once.
You have to install gpsd and configure it so that it listens to the gps device on /dev/ttyUSB0 (the device name is shown in your dmesg output).
see man gpsd
after installing gpsd for configuration information.
Second, it can connect directly to the gps.
When running gpsdrive, click "Preferences" and select the "Settings 2" tab page. At the bottom, you can enable "use serial conn." for the direct connection or diable it for gpsd usage.
When you enable "use direct conn.", you have to enter your device information in the fields below:
/dev/ttyUSB0 as device and baudrate 9600 or 19200, depending on your device.
hope this helps,