Are you sure you're up to soldering two thousand LEDs? At one LED per minute, that's over 34 hours.
The additional HC595s just get put on the end of the existing ones. VCC/GND and pins 11, 12, and 13 are common. See how pin 9 out (SDO=serial data out) goes to the next pin 14 input (SDI=serial data in) and so on. The software that sends the columns out to the 595s through Port D (PD3, 4, 5, and 6) just gets a higher loop count. Since the ATMega32 has a hardware SPI interface, you might want to move the HC595s to those pins (PB4, 5, 6, and 7). If you do this you can replace the author's bit-banged SPI software with the much simpler hardware support. PB5, 6, 7, RESET, and XTAL1 also should be brought out to a serial programming connector. I'm assuming a serial programmer like AVR910 or STK500, so check its documentation for the pin out.
The additional rows will be identical to the existing rows on Port B. Actually, since you have 4 ports on the ATMega32, Port B may not be the best choice. Given other possible uses for many of the I/O pins, I would tentatively consider using PA0-7, PC4-7, and PD4-7 for the 16 rows.
The 24C16B is probably an EEPROM, I'm too lazy at this hour to look it up. The ATMega32's internal EEPROM might make this component unnecessary. You would have to check how much the program uses. If you keep the external EEPROM, the ATMega32 has built in I2C, you might want to move those two signals to PC0 and PC1 which are the ATMega32's I2C pins.
The 10MHz crystal is the MCU's timekeeper. It is the basis for scheduling the LED scanning and setting the baud rate of the UART.
I think it's not time to worry about soldering the ATMega32. I would want to have drawn an updated schematic and taken a first swipe at porting the software. Before you can start on the software you would be installing AVR Studio and WinAVR C development environment (all free). AVR Studio has an emulator so you can try out code one step at a time with no smoke no matter how bad the mistake.
I haven't looked into the program in detail, but the section I previewed appears to have its comments in Finnish. Google Translate probably would make sense of this. Another easy yet tedious task.