How can a tongue-biting parasitic arthropod become a fully functional tongue?
Short answer
There is no evidence whatsoever that the parasitic arthropod Cymothoa exigua develops itself into a functional tongue. Instead, it consumes the fish's tongue and occupies the freed buccal space to continue feeding onto the fish's blood or mucus.  

Background
The linked wikipedia page that states that Cymothoa exigua arthropods form fully functional tongue-replacements refers to Brusca &amp; Gilligan (1983) to back up that claim. While the journal is credible, the article itself is a theoretical work. The authors base their conclusion that the tongue replacement is functional on the relatively healthy condition of just 2 specimens of fish. They reasoned that, because the parasitic arthropods were relatively large, the fish had to be hosting them for quite a while. In turn, given the relatively healthy condition of the fish, the authors reasoned the arthropodic tongues may actually have had functionality. In the figure below taken from Brusca &amp; Gilligan (1983) the parasite does indeed look much like the normal tongue.


Source: Brusca &amp; Gilligan (1983).

However, there is no mention whatsoever that the parasitic tongue taps into the nervous system and I have not found any literature supporting such a claim. Further, Brusca &amp; Gilligan (1983) specifically say that:


  [...] we propose [that] a fish with an 'isopod tongue' [is] not feeding as efficiently  as a non-parasitized fish, [but] feeds more efficiently than a fish with no tongue at all [...] 


The authors' claim is much more nuanced than the blunt conclusion as drawn in the wikipedia page that it is a fully functional tongue; while the effects on its host may be relatively mild, it may not replace all, or even any, physiological functions of the normal tongue. 

Reference
- Brusca and Gilligan, Copeia (1983); 3:813-6