In both cases where people ended up facing such large fines, the RIAA repeatedly offered to settle for much much much smaller amounts, typically around $2-4 per song that the person was sharing.

The defendants decided to reject these offers, even though the MINIMUM POSSIBLE fine if they lost in court would be much larger than the RIAA was asking for.

Typical numbers would look something like this. The person shared a couple thousand songs, and the RIAA asked for maybe $5k. When the person refused, the RIAA would sue over around 20 of the songs, and the RIAA would have an airtight case. The best case realistic outcome for the defendant would be for the jury to award the minimum possible damages: $750/song, so the defendant would end up owing about $15k, which is about 3 times what they could have settled for.

That was their best case outcome. In reality, the jury would generally opt for something between the minimum and maximum, and so they got the big fines.

Then they would appeal. Incredibly, the RIAA would then offer again to settle for a few thousand (total--not per song). And the defendant would refuse, get another trial, and get a jury that opted for even higher up on the damage scale. Oops.

Basically, these defendants were either idiots, or they were taken advantage of by attorneys of questionable ethical judgement who put their desire to argue novel dubious copyright theories in court over the interests of their client.