How does Jesus being nailed to a cross relieve humanity of its sins?

It doesn't relieve us at all. His death, resurrection and ascension opened the gates of Heaven for us, giving us the opportunity to enter with faith. Before Jesus, all souls went to the abyss, Sheol, place of the dead. That's where Jesus descended while He was dead from earth. His death didn't absolve us from our sins. We still have to individually ask for forgiveness every time we sin. that's why the Catholic Church has the blessed sacrament of Confession. Jesus authorized it.