What are the economic benefits of SLAPP in file sharing/piracy?
Background

Since the marginal cost of distribution for a creative work such as a song is now essentially zero, the efficient thing to do would be to provide all songs to listeners at a price of zero (i.e. to allow piracy). However, this neglects the fact that producing creative works involves a fixed up-front cost (e.g. studio recording time, or simply the artist's time composing the work) that cannot be covered if the price is zero. Moreover, it is efficient for producers of 'good' songs (i.e. those that create value in excess of the production cost) to incur the fixed cost, and for producers of bad songs not to do so.

The way that society mostly addresses this issue is through the grant of intellectual property (specifically, copyright) protection. The trade-off made is that artists are allowed to act as monopolist suppliers (with all of the attendant inefficiencies) because the resulting profits will give them an incentive to produce in the first place. The fact that a seller's rewards are tied to their monopoly profits means that the most popular artists should, in principle, have the biggest incentive to create.

Efficiency effect of piracy

If one buys the idea that we need IP protection to induce the production of creative work then tolerating widespread piracy is a bad idea because it takes away the profit that is supposed to provide artists with the incentive to create. Whilst each individual instance of piracy might be efficient, the overall effect of reducing the number of works produced might not. It is then necessary to ensure that enough people do not pirate enough of the time to keep the rewards from production at a sufficient level.

People should be expected to pirate if the benefits from doing do (free consumption of creative works) exceed the costs (determined by likelihood of being prosecuted and the penalty when prosecuted). An obvious way to stop people from pirating is to increase this cost by prosecuting more people—even if you don't really care that much about winning. Indeed, the main objective is not to win the case and get what, to the RIAA is pocket change; rather, the objective is to scare people enough that they start buying rather than pirating music. [n.b. The RIAA would have to win at least some of the cases, otherwise the threat would be completely empty. As Densep notes in a comment, this raises the issue of whether we can really call these cases SLAPPs or not].

If the result is that artists earn more money and respond by producing more creative works then this can indeed be efficient.

Some tangential remarks

The last qualifier (if the result is that artists earn more money and respond by producing more creative works then this can indeed be efficient) is non-trivial. There is some debate about how elastic the supply of creative works really is to the rate of piracy and whether this effect is big enough to offset the efficiency gains that come from consuming creative goods at marginal cost (i.e. zero).

My sense is also that there is a consensus among economists that the existing degree of copyright protection (typically something like the entire life of the author plus an additional seventy years) is far more generous than the level necessary to induce efficient levels of creative goods production.

Lastly, there are other ways of providing an incentive to create. For example, instead of granting an intellectual monopoly, the state can insist that all creative works are freely available and instead reward artists with a lump-sum prize. This avoids the monopoly distortion, but has a couple of drawbacks: (i) the prize would have to be funded somehow (e.g. from taxation, which is itself distortionary); (ii) it would be necessary to find a way to ensure that prizes are only given to reward the right kinds of music. The advantage of the intellectual property approach is that the market automatically rewards popular songs more.