Jesus saves the believer from God's wrath.
God is Holy, and He is angry at us for sinning. This anger is fully justified, for God is just and He cannot be unjust. But God in His mercy sent His Son, that whoever believes in Him will be saved.
We are saved freely, but God cannot leave any sin unpunished, therefor for God to be just in forgiving us, He punished His Son for our sin. Lord Jesus paid the full price for every sin of the believer on the cross (God knows them all as He is outside of time and for Him everything is present).
Once humanity sinned, we became cursed. Jesus took this curse on Himself. Also wages of sin is death, so God provided perfect substitute in His Son, who bore our sins, even though He was without sin, and took them all to the cross, and every believer's sin died on the cross. But because the sinless Jesus died on the cross, the punishment of death was invalid, and God raised Him from the dead as He will do with every believer in Christ.
So Jesus suffered what belonged to us, the sinners, God's just wrath. That's what we are saved from.