How to connect ATTiny13 to USB?
The microbot adapter is a TTL (Typically 5v) based usb-to-serial adaptor. I can't tell what IC they use, but they all act the same. Just hook up the TX, RX, and GND pins to your ATTiny13, and start a regular serial communication. Or in this case, just  ATTiny's TX to the adapter's RX pin and Ground pins together, as the project you link to is transmit only. You would simply replace the max232 in the circuit, with the usb adapter instead. Everything else stays the same.

The max232 is used when converting from TTL serial (0v Low, +5v High) to actual RS232 like a computer serial port would use (+3~25V Low, -3~25V High). Since you are using a usb-to-serial adaptor with TTL levels, the same that the ATTiny13 would use, the max232 is not needed. There are some usb-to-serial adaptors intended to be used with actual rs232 level stuff, so the max232 or similar chip would be useful then, but not in your case.

The 5v from the microbot adapter is directly connected to the usb 5v pin. This gives you up to 500mA to use.