Simple. Aha is a Norwegian band. "Take on me" in Norwegian is, "Ta på meg". "Ta på meg" in Norwegian means, hold onto me or touch me. They caused a stir and left many people to wonder about the meaning when they chose to translate literally, word for word, instead of figuratively. This was certainly intentional as Norwegians are fairly proficient in English.
The phrase "take on me" is nonsense verse that was written to sound cool. It's really just a grammatically incorrect version of "take me on". A-Ha repeating themselves in order to have catchier lyrics.