What are volatility swaps?
Before the introduction of what is now volatility swaps, investors gained exposure to the market's volatility (yes, they already wanted to) through call and put options, products that depend on volatility, but also heavily on the price level of the underlying asset.
A volatility swap is a forward contract on future realized price volatility. Similarly, a variance swap is a forward contract on future realized price variance, variance being the square of volatility. It is basically an agreement between two parties on exchanging $N_{var} (\sigma_{realized}^2 - \sigma_{strike}^2)$ after a given period.
How is the fair value calculated?
A good way to catch the meaning of that fair value is to suppose both actors had to make a consensual prediction on volatility. For example, they expect a realized variance of $\sigma^2$ and they agree on $B$ giving $A$ $N_{var} (\sigma_{realized}^2 - \sigma_{strike}^2)$ at expiry. Then the fair value would be $N_{var} (\sigma^2 - \sigma_{strike}^2)$.
The strike is usually chosen such that the fair value of the swap is zero, i.e. $\sigma_{strike} =\sigma$. This is not always the case, though. This was for the second, simpler question.
How can volatility swaps be used to reduce risk of volatility in the
markets?
Let's take the example of a life insurance company. The company offers many products with guaranteed benefits and these expose them to short volatility positions, that is, they are better off if volatility is low than if it is high. To balance this risk (they are insurance companies after all), they will buy long positions, that is, volatility swaps that gives them green notes if the volatility is higher than they expected. This does not modify their expected outcome, however it contributes to risk reduction. As they are risk-adverse, they may be willing to lower their expected value in order to bring risk down, which mean they are prepared to pay more than the fair price for a volatility swaps, which means other players will be interested in buying them whatever the composition of their portfolios, hence giving birth to the volatility market, which now-a-days has a daily trading volume of about $30-35 million.
Reference: https://www0.gsb.columbia.edu/mygsb/faculty/research/pubfiles/3967/pricing_hedging.pdf
