For example, suppose you stumble across the symbols NEZNIVEHZDHRPNABGEBVNRDHVCEVZHFNOBEVF. Someone tells you that it in fact contains a secret message in ROT13, namely the opening line of Vergil's Aeneid. Should you believe them or are they overfitting?
In this case, it turns out that no system of roughly the same complexity as ROT13 (and that includes any of the 26! examples of 1:1 mappings of the letters) produces anything close to this good a match, so the explanation "this is ROT13" accounts for much more than you would expect by chance. For the same reason, in the linked article the acrostic NIKANDROS in a poem by someone of that name (at least when rendered in Greek) looks too good to be a coincidence.
On the other hand, suppose you encounter the word SXLL and someone tells you that it's the encoding of the word BLUE with a Vigenere square (basically A=0, B=1 etc. modulo 26) and the key RMRH. The math works out, but there are as many degrees of freedom in the key as there are in the supposed codeword (this is a one-time pad, for cryptographers) so you could equally well say that SXLL is an encoding of STOP with the key AEXW. This claim does not pass the point of uniqueness (which for the true one-time pad diverges to infinity of course, that's the point!).
By this technique, most claims of secret messages in Shakespeare fall because there's other, equally complex systems that turn up equally plausible different messages.
In a work the size of Vergil's (or Shakespeare's), you are bound to find some supposed acrostics by chance alone, especially if you allow yourself some degrees of freedom (like skipping the occasional line). My question for the claimed Isaiah one is whether this one is really past the point of uniqueness, something so unlikely under the null hypothesis of coincidence that we can be sure it's an actual "easter egg" hidden here. Of course, the fact that ISAIAH SAYS is on-topic for the plot here is Bayesian evidence in favour of it being real.