It supposedly pulls the skin away from the muscle to allow more blood flow to the muscles for healing.
Who knows if it really works but you look like a boss with it.
It definitely doesn't do that. Basicly Kinesio is just tape that facilitates certain muscles to work more or less depending on where it's placed. The theory is that the touch of the tape activates something called the mechanoreceptors (which are just the cells in our skin that registers touch) which helps active muscles. Ever wondered why someone slaps their thighs and ass before going in for a big squat? You can also use it to physically keep something in position like pulling the knee cap in a certain direction. Kinesio is a nudge though, it's just there to help you keep your limbs in a good position. During the acute part of an injury it might not be that helpful, usually you go with sports tape there.
TS, again it depends on your injury. If you feel like you don't need to go see a physio, which might come back to bite you in the ass, there's been some solid advice in this thread already.
Don't:
Don't work into pain, don't do any bouncing/jumping, fast change of direction or high impact exercises at all for 4-6 weeks. Don't sit on your ass the entire day.
Do:
You can do squats and deadlifts as long as you're not feeling pain and you can control it. Do light mobility work like ankle rotations and movements several times a day, with your foot in the air and with some bodyweight on it as well. Again, don't go into a lot of pain, and if it gets worse give it some rest. Biking or anything else that repeatedly moves the ankle is adviced IF it doesn't hurt. Use a compression band in the first week or two if you feel unstable, keep it for a week or two more if there's any swelling post exercise. Keep your feet up a bit when resting if there's any swelling as well.
Also, start adding in ankle strength and stability exercises in a few weeks. Go slow. You want to build the tissue and stability up to both prevent it from happening again and to be able to tolerate higher impacts down the road.