More precisely, in equity markets, why would one prefer to buy a forward starting option over a vanilla option ? What about the selling side ?
It is hard to imagine why a trader would want to buy a forward start option to express a market view, unless there is a one-off event like an election which they don't want to have as part of the live period for the volatility. A forward start option is mainly exposed to the volatility relating to the period after the strike is set.
Forward start options tend to be used in the creation of "structured products" in several ways. One simple way they can be used is to facilitate a selling period for a structured product. For example, a retail seller will have determined there is appetite for a five year structured product which includes some kind of option. The seller thinks he can sell for example 50 million of the product so can buy 50 million worth of a one month forward start version of the structured product. Then over the one month period they can try and sell the 50 million of product at close to a constant fixed price, as the delta of the product will be very low.
Another use in structure products is as part of a "cliquet" structure. In its simplest form, this would consist of series of for example three month options, each starting when the previous one expires. In a volatile but overall non-trending market such a product can have a better payoff than a simple vanilla option. And all manner of features can be added, such as adding local caps, whereby each option can have a maximum of 10% payout, and/or a global floor in which the minimum payout of the entire structure is fixed.
Such products have somewhat fallen out of favor since the credit crisis due to their complexity. They were particularly popular in the UK and France in the late 1990s and early 2000s.