"This statement is false" - Propositional Logic

In a text I am reading, the section on Propositional Logic says that a proposition is a statement that is either true or false, but not both true and false. Also, from this lecture online, the instructor says that we must be able to associate a truth value to a proposition.

The text I mentioned contains as an example of an assertion that is not a proposition the following:

(1) "this statement is false."

In the margin, the text says that the form of this statement makes it impossible to designate a truth value to it and the instructor in the lecture says simply that, "if [the statement] is true, then it is false, and if it is false, then it is true."

However, why exactly is it impossible to for (1) to have a truth value? What does it mean to say that if (1) is true, it is false, and conversely?

Response to Asaf Karagila
As has been pointed out, I have already asked this question very recently yesterday but it has not received proper attention. This question is one that I feel can be put to rest if only someone would provide an explanation that is direct and suitable for my level, which is that of a novice.

Solution 1:

Let me toss my 2 cents for what your instructor said.

Suppose (1) "this statement is false." holds. Then the assertion inside "" is false. Thus this statement is false does not hold, or (if we abide by the binary logic) this statement is true.

Now suppose (1) "this statement is false." does not hold. Then that statement must be true (as long as we abide by the binary logic.) So the assertion this statement is false is true.

The bottom line is, the statement inside "" does not conform to the binary logic.