Note that I said per song shared, which will be more than the 24 songs that were specifically in the lawsuit. Generally in these cases the person is sharing thousands of songs. The RIAA only names a dozen or so in their suits because there is no point in suing over all of them. They have to prove proper copyright registration for each song and that they are authorized by the copyright owner to sue, so suing over all of them would require a ton of evidence.
With 24 songs, if they win and if they get the MINIMUM POSSIBLE statutory damages (which only happens if the court finds that the infringer was not aware and had no reason to believe they were infringing), that works out to about $5k. If the court finds that the infringer knew or had reason to believe that they were infringing, the minimum statutory damages on 24 songs would be $18k.
So, here's the situation from the point of view of someone that the RIAA catches distributing a few thousand songs: the RIAA will offer to settle for somewhere around maybe $3-5k. That will work out to about $1-2/song. Considering that it is typically $1/song to buy a copy of the song for personal use, being asked to pay $1 or $2 for a copy that you were sharing online is quite reasonable.
If you decide that you are going to make them go to court, and they sue over just a handful of the songs, and if you actually did share the songs and so are guilty and they can prove it, then you are GUARANTEED to have damages of more than what you could have settled for.
It doesn't take a game theorist to figure out that settling is a strictly dominant strategy here.
Note that there is no question of innocence or mistake. Tenenbaum basically admits he did it, that he knew it was illegal, and that he intends to keep doing it. The other defendant who was taking their case to trial (three times now, I believe), Jamie Thomas-Rasset, also was clearly guilty, although she still hasn't admitted it. She did perjure herself, got caught tampering with evidence, and tried to blame her children and boyfriend for her actions.
The RIAA has been sufficiently sloppy that they have pursued these cases in many situations where the facts were against them. In those it does not always make sense to settle.
And, even if the facts are against you, if you can prove that the RIAA screwed up in how they collected that information on you, they might not be allowed to actually present that evidence. The odds of this succeeding are low, but if even one person succeeds in that, the precedent will be good for everyone going forward. (Of course if you fail, you pay full costs.)
Admittedly it takes an unusual kind of person to prefer one of the latter two courses. But if they succeed, it could be very good for everyone else.
Disclaimer, I am not a lawyer, this is not legal advice, and I personally do not care much either way because I have never shared or downloaded copyrighted content in any public forum.