Spiritual: Alma is not explaining the relationship of the Atoning Christ to those who have accepted his name and covenant. In particular, Alma is addressing those who have at least verbally accepted the Atoning Messiah, but whose actions do not follow the gospel of Christ. In verse 20 Alma has indicated that those who do not follow Christ are following the devil, and that this "following" is very specifically related to what we chose to do rather than what we verbally profess.
In the day of judgment, Alma suggests that the atonement is not universal. The only ones "saved" are those whose "garments are washed white." This symbolic washing of the garment uses the garment as a metaphor for the person, and the washing of the garment is the washing of the person. Of course washing is understood and cleansing, and that is the imagery of this verse.
Alma explains that the only way to be washed clear is "through the blood of him of whom it has been spoken by our fathers, who should come to redeem his people from their sins." This statement has two reasons behind it. In addition to the very clear reference to the Atoning Messiah is the reference to the tradition of the fathers. This is important in the context of the times as this difference between the tradition of the Nephite "fathers (prophets)" is precisely one of the points of contention between those of the church and those who believe in the "foreign" religion that denies the Atoning Messiah.
This washing of the garments/person comes through the Atonement, and is specifically symbolized in baptism, which we know was practiced among the Nephites (particularly after Alma the Elder established the church). However, the washing in blood presents another metaphor that should be examined. How does washing in blood cleanse anything?
The salvific power of blood depends upon the sacrifices of the Law of Moses. The blood of the sacrificial lamb had the power to save when painted on the doorposts of Israel in Egypt. Thus this imagery of blood and salvation is combined with the washing of water to complete the symbolic set of images in this verse.
Once again, we must remember that Alma is speaking to the church, and we may therefore expect that these are people who had already been baptized. Thus we must also understand that Alma did not see the baptism as having provided a universal salvation, but rather a temporary cleansing that could be maintained by proper actions, or denied by a return to the un-Christian lifestyle. Of course this is similar to our modern conception of baptism, where it is an entrance ordinance and our lives need to conform to the gospel to continue to have the cleansing effects applied to us.