>Like how is it not an admission of wrongdoing if you finally say "Ok, I'll give you money for this problem to go away".
Because an admission could be used against you in future litigation, thus defeating the purpose of settling in the first place. Settlement functions as a method of allowing litigants to liquidate their conflicts without wasting court resources and expending exorbitant amounts of money going through all steps of the court mandated conflict resolution procedure.
Even if you don't agree that you are at fault, there is some chance that a judge may not agree with you. If you think that chance is 10%, and the suit is worth 100M + you expect to pay 20M in fees, wouldn't it make sense for you to settle for 10M? Regardless of how the case turns out, you're 10M ahead of your anticipated fee spend and 20M ahead of your risk adjusted expected result, and the other side ends up with the risk adjusted value of their claim.