For example, someone who always takes the last piece of pizza without asking if anybody else wants it is "that guy who always takes the last piece of pizza".

The phrase "not to be that guy" is a way to acknowledge that you look like "that guy" (you realize you're doing something that some may find inconvenient or annoying in some way), but for whatever reason, you feel something is necessary anyway in a situation, and you don't take the annoyance or inconvenience lightly.

Continuing the example, if you don't typically take the last piece of pizza, but you you can't ask properly because you're starving and late for a very important meeting, you might say, "Not to be that guy, but I've got to have some food, so I'm taking the last piece of pizza."