Why do you start with var correctGuess = false; ?

It's because the real start of the game happens when we prompt the user for their input. The correctGuess is only ever going to be true or false. This is declared and initialized before we prompt the user for their guess. It would be strange to automatically count their answer as correct before they've even given an answer. (Although, I could say that I've taken history exams where I wish that were the case.) So we assume they're incorrect. However, when they give us an answer then we look at it and if it is correct then we mark it correct.

It's a little like when you're taking a test. There's no teacher I've ever met that's automatically going to mark a blank question as correct just because they assume that if you haven't answered it must be correct.

Hope that makes sense!

Yes, you could do that! But you'd have to rewrite the print statements or call a function to print the winning message instead. This makes our code less DRY. Here's a look at how it'd work without any correctGuess variable at all.

var randomNumber = Math.floor(Math.random() * 6) + 1;
var guess = prompt('I am thinking of a number between 1 and 6. What is it?');
if (parseInt(guess) === randomNumber) { 
  document.write('<p>You guessed the number!</p>');
} else if (parseInt(guess) < randomNumber) { 
  var guessMore = prompt('Try again. The number I am thinking of is more than ' + guess + '.'); 
  if (parseInt(guessMore) === randomNumber) {
    document.write('<p>You guessed the number!</p>');
  }
} else if (parseInt(guess) > randomNumber) {
  var guessLess = prompt('Try again. The number I am thinking of is less than ' + guess + '.');
  if (parseInt(guessLess) === randomNumber) {
    document.write('<p>You guessed the number!</p>');
  }
  else {
  document.write('<p>Sorry. The number was ' + randomNumber + '.</p>');
  }
} 

edited for remove of extraneous comments in code

If we started with this:

var correctGuess;

... the underlying value, in fact, would be undefined. This is a "falsey" value. Here is a list of JavaScript falsey values:
http://stackoverflow.com/questions/19839952/all-falsey-values-in-javascript. Now that being said, it wouldn't be explicitly equal to false. If we later did this comparison:

correctGuess === false;

The evaluation of that comparison would return false.

But the bottom line here is this: there is nothing besides the original declaration of the variable to ever change correctGuess to false. Which means correctGuess will only ever be true if we don't do that. This results in our user always being told they have the correct answer regardless of what they input.

At the bottom of this page at the bottom of "Add an Answer" is a link to a Markdown Cheatsheet that will teach you how to post your code like I post mine

Because as long as you didn't guess the correct number, the correctGuess is false.