"Zecke" is an expression used by xenophobic right-wing extremists. It is named related to the "tick", a small little animal. It is extreme hostile. But I do not know exactly, whether the verb comes directly from this form.
"Anzecken" is slang and no part of the standard language. I agree with
nerven oder
anwidern - but I suppose, it has a rassistic or xenophobic connotation.
In
http://www.forum.nano-aquaristik.de/thread.php?threadid=1735&page=3&sid=4aca8fd7d07cec40079257ef2b1b5b37 it is used more neutral and for animals (small fishes "zecken sich an"), but also in a hostile sense.
May be, it is part of language change.
I asked a friend. He says, it means "
Streit suchen" - to seek a quarrel.