You have it working, it is probably a good idea not to flash anything until you have a better understanding.
Why not just change this one line in his program:
esp8266.begin(9600);
to
esp8266.begin(115200);
and it will work. Then go read about software serial.
As i have mentioned before, I did change it to 115200, and the picture of my serial monitor is the response I got which is not exactly what I expect. It seems that there is still some garbage data printing out





