It's a nominalized adjective in Swedish, those work that way. They don't have the same declension as a normal noun, they only have adjective forms.
An English example of an adjective that works as a noun is 'poor' in "She gives to the poor".
You can think of it like this: there's always a hidden, implied noun after vuxen: "den vuxne" is really always "den vuxne [mannen]" only we don't need to say the noun.
For those who find this word hard to remember, it is cognate with the following verbs, that all mean 'to grow': växa (Swedish), to wax (English), vachsen (German), wassen (Dutch).
Fun fact: 'wassen' is hardly used with this meaning in Dutch any more, only for 'wassende maan' (waxing moon), but it can still be found in the adjective 'volwassen' (adult; literally full-grown).
It is cognate with some other verbs from quite some other Germanic languages, so please, feel free to add :).
See the Wiktionary page on the common ancestor for some more info: http://bit.ly/2axq1TT
Similar to Dutch, "wax" is hardly used in English (American English) anymore, only in referring to the moon, the phrase "waxing poetic," and sometimes the phrase "waxing and waning" applied to things other than the moon. ("Enrollment in this college course waxes and wanes from year to year.")
It still helps me, as an English speaker, to remember the meaning of "växa" and "vuxen." Thank you!