Yes! Game of Thrones fans unite :)

közel = near, close (adverb)

közeli = near, close (adjective)

közeleg = to come, to impend, to get closer (intransitive)

közelít = to get closer (both transitive & intransitive)

It is a verb, meaning "(slowly) getting closer".
"Közel" itself is a .... (I don't know... adjective? adverb?), it means "near", "close by". "Közeleg" is a verb derived from "közel".