Does an annulment wipe away the marriage like it never existed?

No. The Church is really declaring in hindsight that, on the wedding day, specific factors prevented the bride and groom from bringing about a legally valid marriage in church law. The ceremony is not wiped away – all the guests saw it happen! The relationship of husband and wife is not wiped away – that remains the relationship between the man and woman while they lived together. The children are not wiped away – they were born in a loving union, are a blessing from our loving God and remain legitimate in church law.