Hi Mike,
I put ape hangers on my 02 HD Softail Deuce, and chimps (wife's name for smaller ape hangers) on my 06 HD Softail Standard. Ran the wires through the Deuce's bars. Ran the wires and the throttle cable through the Standard's bars.
Main reason? Style. Like RoadKing said, it harkens back to the debut of the Stingray for a lot of us. It looks cool (to some of us), and it feels good.
How does it affect handling? It makes the front end much twitchier - you have much more leverage. When I've ridden them hard, I controlled the bike first with body position and only made minor adjustments with the bars. A little went a long way.
The taller apes reduce circulation to your hands, which is why you only see the young guys using them. The ones at shoulder height don't, and are actually quite comfortable.
I removed the taller apes after a few months because I didn't think they were safe in an emergency situation. I was afraid I'd over-correct. Also decided the stock bars looked better on the Deuce.
Are you thinking of putting them on the Wing? You would be adopted by every HD rider on the road. Really, what one of us needs to do is customize our Wing with a loud exhaust, apes, flames, forward controls, and a suicide shifter. Put on a bandana and ride. LMAO.
There is one more reason for putting them on your bike: to irritate the "squares."

In the same spirit as what Matt from Ashland does to irritate Harley riders: his bike beeps like a truck when he uses the reverse gear. Hilarious!
Rick