I think it's great that you ask these questions. I wouldn't assume that the majority of the community already has a crystal-clear grasp of them since (1) they are not straightforward at all and (2) as far as I know, the answers are not really consolidated in some single post you can read in 5 minutes.
GiveWell's current estimate is $3,000-$5,000 per life saved. This is the range they communicate on their Top Charities page and which they explain here. They have not updated this messaging since November 2020, so that may change soon.
As for a rough overview of the calculation process, this example may help. It is a complex process that starts with the evidence of effectiveness for a particular intervention but then includes a host of factors for which GiveWell calculate and regularly update their best estimates. Some mentioned in the example I just linked to are:
- Not every person who receives a certain treatment/intervention would have otherwise died of the targeted condition.
- Even when an intervention is effective, it probably does not prevent the condition 100% of the time.
- The effects may wane over time for lots of reasons.
- Treatments may not be consistently followed over time.
- Local mortality rates vary across regions.
- Other actors may deliver the intervention if you do not fund the particular program being considered.
If you want to dig deeper, you can go over the cost-effectiveness spreadsheets on this page Michael shared in a previous answer or read this detailed guide.
Your second question was about what "saving a life" actually means. Holden (GiveWell co-founder, now Open Philanthropy's co-CEO) wrote this post about it in 2007. Some snippets:
Approximately 80% of children born in sub-Saharan Africa reach age 5; in the developed world, it’s 99%. [...]
Between ages 5 and 45, people in sub-Saharan Africa have relatively similar mortaility[sic] rates to those in the developed world except for the influence of HIV/AIDS, TB, and mothers dying in childbirth. Around age 45, a lot of the same diseases that kill people under 5 start killing again (maybe due to weakened immune systems). [...]
So what’s a life saved? If you save someone right as they exit infancy (5 years old), you’ve saved someone who probably has around a 50% chance of making it to age 60 … another way of putting this is that if you save two lives [...], you’ve in expectation given one person a full life that they wouldn’t have had.
I will find out whether GiveWell has revisited that more recently.