Litecoin is one of the first altcoins of Bitcoin. It is a nearly identical fork of Bitcoin except with a few parameters changed. All of the following that I am about to discuss work identically on both, so I will just talk about Bitcoin for expediency.
Bitcoin is a digital currency that is based on cryptography, specifically public-key cryptography. The way you spend money through Bitcoin is by using a private key that only you have access to to sign a transaction that sends Bitcoin to someone else. The rest of the network validates that transaction using the corresponding public key and rejects it.
Satoshi Nakamoto wrote Bitcoin. He mined the first block and authored the first transaction. As a consequence of this, several public keys are known that are recorded right into the very beginning of the blockchain. So anyone claiming to be Satoshi Nakamoto should be able to digitally sign a message to that effect with the private keys corresponding to the known public keys, either a transaction on the blockchain or a simple text message.
Anyone who claims to be Satoshi Nakamoto but cannot digitally sign proof to that effect is not Satoshi Nakamoto. The first thing the real Nakamoto would do should he ever come out publicly (which is highly unlikely) would be to provide this incontrovertible proof. As the creator of Bitcoin, he more than anyone knows that you can't take this kind of stuff on faith; it has to be proven mathematically.
The linked post shows the creator of Litecoin demonstrating that he is in possession of the private key corresponding to the genesis block of the Litecoin blockchain. It's really that simple. He has a private key, and he's signed a message that anyone can verify.