How clothes were laundered in Moses' day we can't be sure, but from the very many references in Exodus to such washing, we can safely conclude that it was a commandment of the Lord. When the children of Israel came to the Wilderness of Sinai, we read: "So Moses went down from the mountain to the people and sanctified the people, and they washed their clothes." (Exodus 19:14) Clean clothes were a required part of consecration toward a holy God.
The New Testament writer of Hebrews urges us to "draw near to God with a true heart in full assurance of faith . . . and our bodies washed with pure water." (10:22) When Jesus washed the feet of His disciples, more transpired than the removal of road dust. His example of humility showed the way of putting love into action, a willingness to help others.
The scene shifts into heaven where the Apostle John is shown a great multitude of people from all nations, tribes and tongues. They are arrayed in white robes. One of the elders who stand before God's throne identifies them, saying: "These are the ones who come out of the great tribulation, and washed their robes and made them white in the blood of the Lamb." Revelation 7:14 NKJV
Cleanliness may not actually be next to godliness, but it pictures a greater cleansing, one of which John testifies in Revelation 1:5: "To Him who loved us and washed us from our sins in His own blood . . . be glory and dominion forever and ever. Amen."