Arduino AVR Boards is a package of software that allows the Arduino IDE to program the Arduino boards of the AVR architecture, such as Uno, Mega, Leonardo. There are also other packages you can install to add support for other architectures, such as the SAMD architecture used by the Arduino Zero and MKR boards.
Because the Arduino IDE comes with a bundled version of Arduino AVR Boards, we're used to associating a particular version of Arduino AVR Boards with each version of the Arduino IDE. However, the Boards Manager feature now allows us to use any version of Arduino AVR Boards with any version of the Arduino IDE. The author of the code may have been using the old IDE version with a new Arduino AVR Boards version. You could do the same if you wanted. The reason I recommended updating the IDE is because there have been some isolated issues where a particular Arduino AVR Boards version had issues with some older IDE versions. I have no idea whether Arduino AVR Boards 1.6.23 will have problems with Arduino IDE 1.6.5. What I do know is that Arduino IDE 1.8.9 works fine with Arduino AVR Boards 1.6.23. So it seemed like the safest advice. However, you could likely continue to use Arduino IDE 1.6.5 if you prefer.