According to Lutheran teachings (Martin Luther, Lutheran confessions, any Lutheran Church, Lutheran theologians, etc.), why did Jesus Christ had to die on the cross, and what did this accomplished?
why did Jesus Christ had to die on the cross
Because Adam sinned and damaged our relationship with God. Something needed to be done to restore that relationship, and nothing we humans could do would suffice.
what did this accomplished?
Salvation. Through Christ, we are reconciled to God and receive eternal Life through belief in Him. "I am the way, and the truth, and the life. No one comes to the Father except through me." (John 14:6) Christ is the propitiation (atonement) for our sins (1 John 2:2).
Some articles for further reading: