Nobody knows. Probably because it's handy, and Germans like it handy, and if the handiness is American, it is a safe bet that Germans love it and use it to death and to the confusion of everyone else. We are a strange people.
According to Pfeifer, there are probably two origins. The one quis_lib_duo mentioned and there was a World War II era handheld two-way radio called "Handie-Talkie", which was manufactured by Motorola and used by the US military. So I guess it's not that strange after all. It just causes some confusion because many Germans think "Handy" is the regular English word for cellphone, and they're not aware of its lewd meaning when used as a noun in English :)
Interesting, though I'd aver that even most native English speakers aren't aware of "its lewd meaning" per se (i.e. I don't think it's a particularly common synonym for what I take it you're suggesting), so I don't think "Handy" would strike most English speakers as anything other than a quaint appropriation of an English adjective, in the context of German.
I could simply be out of the loop, but this seems more like an expat joke than anything. In other words, in a context where the word does get assumed into English, and an English speaker then has to guess what this nominalized form means in an English context, it's easy to come to the wrong conclusion, so to speak.
PeaceJoyPancakes wrote I'd aver that even most native English speakers aren't aware of "its lewd meaning"
I agree. My husband and I are both native English speakers, and neither of us was aware of a lewd (or sexual) meaning. To us "handy" just means useful or convenient, etc. I looked it up at www.merriam-webster.com, and did not see a lewd meaning. But, I did look it up at the "urban dictionary" and found out what you are referencing.
By the way, in the US you can say cell phone or you can just say cell. Both are fine.
I think it's because you can have it in your hand and carry it around. Believe me, Germans don't necessarily know what "handy" actually means. We also say "Smartphone" or "Mobiltelefon" if it makes you feel any better ;-)
There's a logic in germans using the word " handy" because it is handy. The English use the words " mobile phone" because it is mobile. What's the logic to call it a cell phone? Is it because prisoners use them ?