"Volatility" in the term "Volatility Trading" does not mean the stock's movements, it is a way of measuring the excess value in an option beyond what the parameters of the option would imply. That excess value is usually referred to as the market's assumption about the future volatility of the stock, but really its just an error term influenced by market participants based on supply and demand. Low volatility means "pretty close to its theoretical value assuming no volatility" or to put it another way: "cheap" i.e. good for buyers and bad for sellers.
Sort of like how different companies with the same cash flows can trade at different multiples, otherwise identical options in two companies (or different expirations/strikes in the same company) can trade at different prices because of the opinions of market participants. Volatility traders act when the different prices/error terms are too far apart, counting on the prices/error terms to converge a.l.a. pairs trading.
Since they are trading the error term directly, they attempt to construct positions that remain relatively flat in value as the stock moves around, but are designed to only change in value when the error term changes. That is how they can make money "both ways", because they can profit if the stock goes up, down, or stays the same, as long as the error term moves in the correct direction.
The reason you only see sophisticated people doing this kind of trading is because you need a large and complex position with many hundreds of options to be in a truly market-neutral environment. You can't take advantage of mispricing without such a large position because buying/selling single options involves a tremendous amount of risk, so you need to do that as a part of a larger portfolio to spread that risk. Retail traders tend to spread the risk by doing 2 transactions (the mispriced option and a well-priced but mirrored hedge option), but that is a) much more expensive from a commissions standpoint and b) really limits the range of market-neutrality forcing you to adjust more frequently to stay market-neutral, again, with commission costs.