But this is not an ordinary
tautology, such as "all three-legged black cats are black and have three legs". The
truth conditions for this
sentence are very odd, and it in fact fails to be
meaningful for the same reason that "This sentence is false" does, namely that it is
self-referential, and the
reference cannot be
resolved into conditions that can be checked.
Something like "the sentence on this card has nine words" can be self-referential but checkable. It is a physical thing. Truth however is not a property of the physical inscription or utterance, but of the logical statement, and it is that that is unresolvable: what statement? The statement that the statement that... ad infinitum.
The technique you begin with in both cases is to say: "Let's assume it's true. What happens? Now let's assume it's false. What happens?" With a meaningful sentence you might expect one assumption to be consistent and the other to lead to a contradiction, so you could confirm that the consistent assumption is the correct one.
In the case of "This sentence is false", however, both assumptions lead to a contradiction. As Tosta Dojen has pointed out, in the case of "This sentence is true", both assumptions confirm the assumption. This does not prove it to be true. In this case consistency is not sufficient for truth.
With both sentences, "assume it's true" and "assume it's false" fail to resolve its truth. So the remaining alternative is that each of these sentences is neither true not false. They look, especially this one, grammatically meaningful, but you have established logically that they can't have any real meaning. The sense seems to be there, but it cannot be given a reference.
Fuzzled thanks to funnytoes for pointing out that my original addition was not different enough from Tosta Dojen's.