You haven't told us which country you live in, or what this wire connects to. (It makes a difference if it's say, a refrigerator or microwave oven, as opposed to a small room fan.) Still, if it's connected to your mains (115/230V) power, as others have mentioned, your friend was right: that wire's a serious accident waiting to happen.
In addition to the excellent answers you've already received, let me add a couple things that I hope will be relevant not only to you, but others in similar situations:
Whenever I find myself pondering things like the following, a sanity check is surely needed:
My friend claims that this is highly dangerous, but I don't think that touching it would be lethal
I think an important test is that you've a) been told it's dangerous by someone who presumably cares about you, b) you have willfully rejected that advice, yet c) you haven't actually tried touching it yet! (DON'T try touching it. Please. Possibilities range from no sensation, to a light tingle, to cardiac arrest.) I'm glad you've come here for advice, and please consider getting your friend some flowers or something.
Mostly country-agnostic practical advice
Step 1. Unless it's connected to some actual life-support equipment or other vital systems, unplug that cord now, then cut the plug off so nobody else plugs it back in.
Now that your life and home are no longer in danger, you have a few options (subject to local law/regulation of course):
Check the device. If you're lucky, the entire cord may be removable from the back of the machine, making replacement cheap, easy, and safe.
Replace the entire device and recycle the old one. Especially if the device was inexpensive, and if you are not skilled with wiring, this may be a very reasonable option.
If permitted by local law, you can cut out the damaged section of cord and re-attach the plug (or attach a new plug if the existing one is damaged—the plug is not visible in your picture, so we can't tell). If the damage to the cord is too close to the device itself (say, less than a metre), the entire cord may need to be replaced.
For an experienced electrician, either repair can be done in a minute or two with basic tools. There's plenty of info online, but I would suggest you check with a local electrician or home wiring adviser in your local hardware store. Hooking up a plug is easy enough, but you want to make sure you're not violating any local regulations. If your "fix" starts a fire, your home insurance might refuse to pay, and in some jurisdictions you might even be criminally liable.
If you perform any kind of repair, verify that repair with a multimeter. Some devices will operate with improper grounding, continuity, or poor connections, but may still be unsafe.