Why should I prefer tea dust over crumbled tea leaves at home? I just noticed that the red coloured packet of tea contains the tea dust! 

This is very surprising. Tea dust available in that kind of packet means that it is available for home use, otherwise it would have been in a tea bag. For commercial use, I don't think selling such small packets makes sense.

How am I supposed to use this tea dust at home? Why should I prefer it over the normal crumbled leaves of tea?
You don't necessarily want tea dust. The reason commercial producers grind it so fine is to maximize the flavor output and steep speed with the least amount of tea, but and this is my opinion only, I think that makes an inferior tea.

Good tea just uses more leaf so it can reach the desired richness in less than 5 minutes.

Steeping for longer draws more tannins out of the leaf and woody bits and makes tea more astringent, and so does grinding the tea too finely.