The crux of the paradox is that it allows grammatically and semantically correct sentences to be formulated that cannot consistently be assigned a truth value.
If we assume 'This sentence is false' is true, then the statement must false.
If we assume 'This sentence is false' is false, it must be true.
In either case logic leads to a contradiction since we must conclude the statement is both true and false, leading to a self-referential a paradox that cannot be resolved.
While it may seem trivial, solving the Liar Paradox is important in understanding the philosophical concept of truth. That is developing a theory of truth or a definition of truth or a proper analysis of the concept of truth.
To date no universally accepted solution has been found, although no doubt some would argue that politicians solved this paradox years ago!
The five most common solutions that have been put forward are:
The sentence is ungrammatical and has no truth value.
The Liar Sentence is grammatical, but meaningless and has no truth value.
The Liar Sentence is grammatical and meaningful, but has no truth value.
The Liar Sentence is grammatical, meaningful and has truth value, but one other step in the Paradox is faulty.
The argument of the Paradox is acceptable, and we need to learn how to live with it being both true and false.
There's a wealth of material on these available on the internet, for those who are interested, but what I find even more fascinating is that maths has its own version of the Liar Paradox.
In the 1930's Kurt Gödel proved that some mathematical statements can never be proved true or false. In maths you can of course change the Axioms (base assumptions) on which proofs are built to solve a problem, but that still leaves other problems unsolvable.
Thus there are no universal Axioms that let you prove everything. The standard axioms used in modern maths are known as the 'Zermelo-Fraenkel set theory with the axiom of choice' or ZFC for short.
In 1936 code breaking genius Alan Turing took this a step further, he developed the hypothetical 'Turing Machine', a mechanical device that could mimic computer algorithms. He also proved that using the standard axioms (ZFC) there must be some Turing machines whose behaviour cannot be predicted.
Two researchers from the Massachusetts Institute of Technology have recently simulated a Turing Machine on a computer to test this theory. This should only stop cycling through the 7918 states of the ZFC axioms if they are proved false. While this won't resolve the 'Liar Paradox', mathematicians will simply fall back on another set of axioms, the race is now on to find the fewest states required to create a paradox.
So this sentence is still false.