Theory 1: moisture
Were the leaves wet at all? I worked in a restaurant before, tending to the fridge. I had to make sure the salads were washed and thoroughly spun. Wet greens wilted and rotted faster and we had to toss them. The roots can be wet and exposed to water, but make sure the leaves stay dry.
Theory 2: freezer burn
This link mentions that cilantro can be damaged by freezing temperatures, which could be an explanation for what you're seeing? Is it so cold in your area that your refrigerator actually dips into freezing temps because the air around it does?
Meats can get freezer burn or taste off after freezing. One explanation I've read is that water freezes as sharp crystals in and around the cells. Upon defrosting, these sharp crystals can cut through cell membranes, causing cell contents to leak out and alter the meat structure. In theory the same could apply to anything organic.
Theory 3: dehydration
The link above and others I've read suggest storing cilantro as you do in the fridge, but covered with a plastic bag. Some wrap their cilantro in a paper towel. Either way, this creates an enclosed or somewhat-enclosed space that slows down moisture escaping from the leaves. The air in the fridge is generally dry and can suck moisture out of the leaves. I know this sounds like the opposite of too much moisture, but there's a window for how much moisture plants want--neither too little nor too much is desired.
I'd try to resolve this in the order above. If you left the leaves wet, I'd say that's a very likely culprit. I'd check your fridge temperature next: you could put an ice tray near where you put the cilantro to see if it freezes or starts freezing overnight. If it's still wilting fast, try doing what you did with a plastic or paper bag over the leaves--making sure the bag isn't crushing them.
I am not aware of any method of freezing herbs that will maintain their freshness, in the general case, especially for delicate herbs such as basil.
The best method I know of to freeze herbs, which I personally would not do and have never tried, is to mince or puree the herbs, then freeze them with water. You might do the freezing in an ice cube tray to have small quantities available. The water is to help prevent the volatile flavors from evaporating, which will happen slowly even at freezer temperatures. Of course, this is still very different than fresh.
I will try to answer the alternate question which is probably what you really want to know: what is the best way to hold fresh herbs to maintain their quality?
The best way to hold fresh herbs is to treat them like a bouquet of flowers. Trim the bottom of the stem ends, so there is a fresh, clean cut. Then put them in a small drinking glass, measuring cup, or other container with water, much like you would put flowers in a vase. The entire thing can then be put in the refrigerator (florist storage rooms are refrigerated, too!). With this technique, you will often be able to get a week or more. (BTW, it also works wonders with celery, which is essentially a giant big herb.)
Probably the second best way is to layer the herbs in a single layer in paper towels, then roll or fold that up so you can put it in a zip lock bag that is not completely sealed (you don't want the moisture from their respiration--herbs are still alive on the stem--to be stuck inside). This too would go into the fridge. You should get at least a few days with this method.
With either method, you can wash the herbs before storage, but with the paper towel method, they should be thoroughly dried after being washed.
Best Answer
Assuming the jar is in the fridge?
Fridge is very drying and the small amount of water sucked up from stems (that leave the water to be cut) isn't enough in your case. Cutting while remaining underwater sometimes makes a difference.
I have better luck wrapping herbs loosely in very damp papertowel and storing in tupperware in fridge. After 2 weeks even fragile cilantro lasted with only a daily picking-off of a dark leaf or 2.