The button for the zeroing has nothing to do with the LCD, it is just a button tied to a digital pin (in this case D0) which has the pull up resistor enabled, when you push the button that digital pin is put to the ground and this condition is tested in this section of code:
if(!digitalRead(BUTTON_PIN)){
lcd.clear();
lcd.print("Zeroing...");
esrCal = (miliVolt)/current;
lcd.print(" done!");
lcd.setCursor(0,1);
//writing calibration value into EEPROM so we don't have to calibrate on restart
eeprom_write_block((const void*)&esrCal, (void*)0, sizeof(esrCal));
lcd.print("saved to EEPROM");
delay(400);
}
I suppose you have some kind of serial interface LCD so you must set up your LCD to use a proper library, the schematic here shows HD44780 type LCD wiring using 4bit mode, more on LCD you can find here: Arduino Playground - LCD