There is no serial port on the at tiny88. Follow the instructions - click upload in Arduino IDE while the board is unplugged, and the downloader will request you to plug in the USB when ready. Instead of unplug-plug sequence, you can also just hit the RESET button on the device when requested to plug in the device.
It's really a pitty that the USB device that takes care of programming the chip on the board does not provide a virtual serial port - even if we had to use a software serial emulation in tiny88, this would have made the board much more useful.
But you can still use software serial library and use some digital pins and communicate with other devices over serial protocol, or with a computer if you have a separate USB-serial converter, which you can get for a very friendly price on-line, thus making it still a half-price solution as compared to Arduino.