The last of the priestly garments was the most private. The Targum Pseudo-Jonathan on Exodus 28:43 explains that Aaron and his sons had to wear the fine linen undergarments — the mikhnasayim — whenever they entered the tent of testimony or approached the altar. Without them, the Targum warns bluntly, they would receive the punishment of flaming fire.
The detail is not modesty for modesty's sake. The altar had a ramp, not steps, precisely so the priests could ascend without exposing themselves. The breeches doubled the protection. Holiness, in this vocabulary, is not prudish, but it is particular. A priest standing higher than the assembly, handling fire, dealing with blood, needed to be covered so that the service did not accidentally become spectacle.
The Targum frames the undergarment as a life-or-death matter, and the Sages heard that seriously. The smallest garment, hidden under every other layer, was the one that decided whether a priest lived through his shift. Greatness rests on invisible supports. The public vestments dazzled; the hidden linen kept everyone alive.
The verse closes with the phrase an everlasting statute for him and for his sons after him. Whatever else changed between the Tabernacle and the Beit HaMikdash, this rule did not. The first garment a priest put on was the one no one would ever see.