I see in the picture that there are only 2 phone jacks on the card. How does the card utilize the 8 modems simultaneously? How does the wiring work? How many analog lines would I be using?

The card comes with a cable that converts one RJ45 to four RJ11 jacks.

Each modem requires one analog line. For four modems, you will need four analog lines.

All the modems will show up as separate modems in the Windows or Linux OS, it is up to the application to make use of all the modems simultaneously. Your application will need to support this.