What does refrigerating do to fermentation?

I've made root beer a couple of times using champagne yeast. It got enough carbonation after a few days. I'd put it in the refrigerator to stop the fermentation. I've used plastic bottles only, and I've had one squirt all over the refrigerator.

So now I have a plastic bottle of half done apple cider. It was left over from some 5 gallon bottles so I thought I'd cap it off and have a carbonated apple drink. The thing is, it doesn't seem to stop fermenting. I've bled off a lot of gas several times, but it just keeps coming.

You probably sucked up too much yeast. Especially if that was the tail end of your batch. I move my primary to a spot where I will transfer and leave it there for at least a few hours before I transfer/rack. If it has fermented in a cool environment, your best bet is to keep it in that temp and rack at that temp. If you move it to a warmer spot, it's impossible to rack "carefully".

What I am curious about is, does making the yeast cold (as in the fridge), kill it? Simply stop it altogether? Stop it until it warms up again?

It doesn't kill the yeast, just puts it in a dormant state. If it is taken out of the fridge, has residual sugars to eat, and left in a warm place, it will start fermenting again. Pasteurization is an option if you wanted to store it out of the fridge, and there is a great thread about it on this forum.

Well it aint dormant even though it's in the fridge. Maybe it's got infected or something. I let a little bit squirt out and when I smelled it, there was a strong sharpness to it, like vinegar maybe. Tasting it I can't really tell if it's sour or vinegary.