You are correct and incorrect in your thinking. Yes the two need to be connected to a network. Specifically they need to be connected to a network that supports IP addressing. However, that network can consist of a single cable between your PC and the board.
1 - Grab an extra network card and plug it into your PC.
2 - Use the OS to configure your new network interface to use Static IP addressing and give it an IP address and set the subnet accordingly.
3 - Set the Simple Socket Server example code to use static IP instead of DHCP and set the IP address and subnet accordingly.
4 - Connect your new network interface on your PC to the network interface on the FPGA board. You may need a crossover cable or adapter. Many PHYs now have crossover built in. If your network doesn't show a connection then you need a crossover.
Jake