This fallacy arises from the incorrect application of the rules linking orthography to phonology1, resulting in an argument that 'ghoti' should be pronounced similarly to 'fish':
gh, pronounced [f] as in enough [ɪˈnʌf] or tough [tʌf];
o, pronounced [ɪ] as in women [ˈwɪmɪn]; and
ti, pronounced [ʃ] as in nation [ˈneɪ̯ʃən] or motion [ˈməʊʃən].
This is not a definitive list.
Key to the phenomenon is that the pronunciations of the constructed word's three parts are inconsistent with how they would be pronounced in those placements. To illustrate: gh can only resemble f when following the letters ou / au at the end of certain morphemes ("cough", "laugh"), while ti can only resemble sh when followed by the letters -on / -al ("confidential", "spatial") etc.
Also of interest is that at the time (around mid-1800s) many persons were intrigued with the 'Phonotypy and Phonography' of English, with Alexander J. Ellis presenting a number of absurd respellings, like turning scissors into 'schiesourrhce' by combining parts of schism, sieve, as, honour, myrrh and sacrifice. (Sacrifice was historically pronounced with a [z])
Source: Wikipedia, NY Times
1 Thanks to Azor-Ahai for confirming this. I originally thought it was an error with phonemes.