The secret is that they are dabbing their index and/or middle finger in the a little pot like this one. These usually contain cotton wool which has some kind of oil soaked in to it.
Why do we do this?
Well, strings can become sticky and friction can build up between the string and your finger (which should have callouses), making it difficult to slide up and down the string at speed. So, we use some light mineral oil, just a touch, to make the glide up the string fast and smooth.
What kind of oils to use?
This is purely up to you. A popular choice is coconut oil. It smells lovely and stays smooth after a while of practising. It also has a slight drying effect on the skin so your callouses stay hard. The downside to coconut oil is that it can become a bit gummy and sometimes even go hard if you don't wipe it off after you play. It can also make the lines in your fingers go black, I'm not sure why, perhaps it is reacting with the metal from the frets and string.
Some people like Baby Oil but I don't recommend this. While is it a nice oil (smells good, is easily available and to start with it is smooth to slide with), it tends to become sticky after a while of playing and can actually have the opposite effect from what you wanted in the first place, making your finger stick to the stick and even increasing the friction.
There are loads of oils to try. I've heard mustard oil works well, as does light olive oil. As long as it is a light mineral oil it should work well.
Just don't forget to wipe your strings down after you play to prevent build up of gunk!
For the video today I'm posting Shivkumar Sharma. He is a massive name in Indian music and plays the santoor. He is playing with another legend, Zakir Hussain. Zakir is one of the greatest tabla players in the world. They are playing the beautiful raga Kirwani.