wire
I have this wire some of the inner part of which is exposed. My friend claims that this is highly dangerous, but I don't think that touching it would be lethal, and in the worst case scenario there might be a short circuit, though the plug is grounded (3-prongs). I don't understand enough to know this myself…
Garage door sensors and buttons are low voltage (usually 24vdc), and low enough current that for the short runs typical in a garage you don't need very large cables.
18awg "thermostat wire" will work, but you can go even cheaper with 22awg wire, usually sold as "security system" or "central vac" wiring (2 conductor), or telephone station wire (4 conductor - though you only hook up two).
Also, a little bit of heat shrink tubing goes a long way to making a neat and professional looking splice.
Those are indeed hazardous "exposed wires", and you should take immediate precautions to make it impossible for anyone to inadvertently touch them. Ideally, you'd turn off power to that circuit and either keep it turned off, or while the power is off cover the exposed wires securely with an insulating device (a securely fastened wire nut, electrical tape, etc) until you can have someone replace the light fixture.
Best Answer
Given the colours (I don't know why the other answerer referred to a red wire, the wire in your picture is clearly brown) I'm going to assume you are somewhere in Europe.
The insulation on the nominally live wire is gone, that means you more likely than not have 230V exposed (if you are in a country that cares about polairty you definately have 230V exposed, if you are in a country that doesn't care about polarity it's 50/50). Touching that is very likely to result in an electric shock, if you are unlucky said shock may be fatal.
There is also earth exposed. With some movement you could end up with a short circuit. Most likely this will blow a fuse/trip a breaker but if you are unlucky it could also start a fire.
I would advise against using insulation tape to repair mains wires except in a dire emergency. It insulates fine but the adhesive can lose stickiness over time. If you do insist on using it make sure you remove the outer sheath, insulate the damaged wires individually and then insulate the bundle.
Best option is to replace the cable, if that is not possible then it may be possible to cut it and insert a suitable joint.