Is it true that Mongol warriors wore a layer of silk clothing underneath their armour because arrows cannot easily cut through silk, and thus they can easily pull out any arrows just by pulling out the silk?

I find this to be highly suspect as an explanation, least of all because an arrow can cut through silk. But, nevertheless can think of a reason why silk would be popular, and part of that reason might be a misunderstanding of the reason you just gave. I can't speak directly for the Mongols, but in the 18th and 19th century, silk shirts were very, very popular for dueling attire. This was for two reasons.

First, silk is a very durable material compared to other common clothing constructs like wool or cotton, and if you were shot or stabbed, it was less likely to tear into tiny little shreds that would get stuck in the wound. Instead, it would probably be only one or two pieces, so quite easy to remove (I suspect this is what you may be thinking of. You wouldn't be able to pull the arrow out just by tugging the silk, but you would have a very easy time cleaning the wound of debris. I've been racking my brain to remember what movie it is, but there is a scene in... some movie... where a doctor is pulling pieces of cloth out of a wound, and them comparing the pieces to the hole in the shirt to see if he has it all. It is illustrative of what I'm talking about here. Less pieces makes this easier! ~~Might have been one of the Sharpe films?~~ It was Master and Commander. Thanks /u/bearcare101)

Second, although this obviously wasn't well understood before germ theory came about, silk has a notably high antibacterial quality, and thus debris that gets into the wound would be less likely to cause an infection. The reason for this wouldn't have been understood my the Mongols of course, but the effects might have been observed.

So again, this isn't a reason the Mongols specifically would have worn silk, but it is a reason why someone, generally, who is expecting to receive a combat wound, might consider it.

TL;DR: An arrow can cut through silk, but it is less likely to cut silk into many little pieces.