I watched a YouTube music tutorial on guitar bending exercises which stated that while bending on guitar the first finger has to be placed on the top strings for muting all those other than the strings to be played on.
As an example it suggests that if I want to bend the G string then the first finger has to be used for muting the A and D strings, whereas some videos didn't speak anything about it.
Now my question is: do I need to do this?
I would not say that the first finger must be used to mute the lower strings but you can do so if you wish. There might be a particular circumstance (trying to achieve a certain effect perhaps) where muting the lower strings with your first finger might help.
If you are picking only the string you are bending at the time the note is played - it is usually not necessary to mute any adjacent strings. In fact when doing full bends or greater, many guitarists will use their first finger to aid their middle finger in pushing up on the target string (the one you are bending).
If you edit your answer to include a link to the YouTube vid you mentioned, it might be easier to give you a more specific answer.
The muting or damping you mention is not related to bending or any other technique. If you want to ensure no unwanted ringing on strings other than those you are trying to play, then muting can be very useful, but whether or not you want to at any specific time is up to you.
I sometimes mute (but not necessarily with my left hand - sometimes it is more appropriate to use your right) and sometimes don't. Sometimes I need to in order to avoid catching the next string up and sounding it - but other times it doesn't matter.
And as @RockinCowboy said, bends may use 1, 2 or even 3 fingers as needed.
So don't be forced into thinking one technique is the way to do things - try them all and see what works for you at the time.
When you are bending strings your fingers catch other adjacent strings in the direction of the bend, so if you're amplified you can usually hear those strings vibrating, and when you release the bend, if the ajdacents strings aren't properly muted, they will ring out.
There are two approaches to muting the strings: Fretting hand muting, where you are muting the adjacent strings with your index - this, however, means that your index isn't going to help the other fingers on the string that's being bent. The other approach is picking hand muting, where you are using your picking hand palm to mute any adjacent strings (if lower in pitch) or your remaining fingers (middle, ring and pinky) to mute the rest of the strings (higher in pitch).
I find the picking hand approach more suitable for muting the strings that I don't want to ring, but you can use whichever feels more natural.
