According to Reynolds and Sjodahl, the expressions "their garments are made white in his blood" (1 Nephi 12:10) and "these are made white in the blood of the Lamb" (1 Nephi 12:11) are also found in the Revelation of John (Revelation 7:14), where the angel explains that the multitudes standing before the throne, and before the Lamb, arrayed in white robes, are they which came out of great tribulation and now are serving God day and night in his temple. They, too, "have washed their robes and made them white in the blood of the Lamb." Therefore they are before the throne of God.
This exceedingly solemn imagery recalls to our minds the sacrifices of the pre-Christian dispensations, and particularly the sacrificial rite on the Day of Atonement under the Mosaic law. That was a day of fasting, confession of sins and atonement, followed by rejoicing.
On that day Aaron, or whoever held the office of high priest, clean and attired in priestly robes, came to the Sanctuary with a young bullock for a sin offering and a ram for a burnt offering. These he offered and thus made atonement for himself and his house. Some of the blood he sprinkled upon the mercy seat, i.e. the covering of the ark in the Holy of Holies. He was now prepared to officiate as mediator between God and the people.
In this office he presented two goats before the Lord at the door of the tabernacle of the congregation. By lot one was selected to be the Lord's; the other to be the "scapegoat." The high priest now laid his hands on the scapegoat and thereby transferred symbolically the sins of the people upon him. The animal dedicated to the Lord was slain as a sin offering. The scapegoat was led into the wilderness and set free. And thus an atonement was made for the Priest and the sanctuary and for the entire people. According to the Talmud, in the evening the maidens all went forth arrayed in white garments, into the vineyards, where they rejoiced, singing and dancing, and inviting the young men to come and select their brides. They had, figuratively speaking, washed their clothes and made them white in the blood of the lamb, the prototype of our Lord, the Lamb of God. Their sins had been confessed, atoned for and forgiven. . . . [George Reynolds and Janne M. Sjodahl, Commentary on the Book of Mormon, Vol. 1, pp. 108-109]