I first heard this euphemism back in 1997 and assumed it was a southern thing, I'm apparently wrong. It has spread from whence it came and is pretty much everywhere, even in the news. The scenario that I heard it in went as follows:
Sarah: Awwwdry, have you seen a picture of Brenden yet?
Me: Who? Oh yeah...the blind date. Nope, why does Amanda have a picture for me now?
Sarah: Yes. You simply must see him, he's hot as balls.
Me: I'm sorry...he's what?
Sarah: Hot as balls, I kid you not.
Me: I'm going to assume that's good..where's the picture?
In the form that my dear friend Sarah presents it in, the phrase means good looking. Not just your every day handsome boy, but drop dead gorgeous. They simple do not get hotter than "hot as balls".
It's a pretty nasty picture to put with good looking. I'm guessing that back when the whole "tsst, I am so hot" thing was popular this evolved from that. It is common knowledge that testicles get a bit heated as the groin is one of the warmest place on the human body in either sex. In fact, if you are stuck in the cold it is sometimes recommended that you put your hands either in your armpits or your groin to prevent frozen fingers. The scrotum is designed to keep the testes at several degrees BELOW normal body temperature, since sperm production decreases the higher the temperature gets.* My search for the origin of this phrase came up empty but I did see it used in a different way and even came across clothing sporting it.
G-Wear, a clothing site for the gay community and it's friends, has a t-shirt for sale on their online store with this phrase on it. I'm going to make a leap here and guess that they are using it in the afore mentioned context, saying that the wearer of the shirt is in fact quite attractive. If you think you or a friend are "hot as balls" feel free to spend your $17 at the link below. Incidentally on the page it says "it's a New York thing" perhaps this is a clue as to where the phrase originated from?
The other context that I've seen it used in is in regards to the actual temperature. (Again, the groin is one of the warmest places on the body.) Junkies Radio broadcasting from WHFS 99.1 FM Studios in Lanham, Maryland held a contest titled the "Hot as Balls Marathon" as a ticket giveaway. They took five contestants, made them put on enough layers to survive a Canadian winter, and had them run a race in 106F weather. These people would have indeed been pretty hot in the already sweltering temperatures.
And finally I came across an actual temperature for "hot as balls" at Pseudodictionary. There, under the definition for balls, it says "'Hot as balls' is around 100 degrees Fahrenheit." Good to know.
References:
* Thank you gwenllian for helping me keep things clear.
G-Wear, http://www.gwearonline.com/hotasballs.html
Junkies Radio, http://www.junkiesradio.com/gallery/2002hotasballs
Pseudodictionary, http://www.pseudodictionary.com/search.php?letter=b&browsestart=80 | 
