Which clothes is the Pasuk referring to?
Rashi (in Devarim 32:50) and Ramban (citing the Toras Kohanim): He (miraculously) removed the Bigdei Kehunah Gedolah from Aharon 3 and placed them on Elazar, garment by garment, in the correct order, in front of Aharon.
Ramban: Aharon ascended the mountain still dressed in his Bigdei Kehunah after bringing the Korban Tamid and the Ketores. Then Moshe removed them and placed them on Elazar after dressing Aharon with the shrouds that he had prepared for him.
See Sifsei Chachamim.
Thereby inaugurating him as Kohen Gadol. See also Ba'al ha'Turim.
Ramban: After Hashem had dressed him with Bigdei Shechinah underneath. See Ramban.
Seforno: Leaving him dressed in the four garments that he wore when he entered the Kodesh Kodshim on Yom Kipur - in the way that the Mal'achei Elokim appear to Hashem's servants.
This is unlike Rebbi (Yuma 23b), who says that the Avnet of a Kohen Hedyot is unlike that of a Kohen Gadol. (PF)
How could Aharon wear his garments there? One may not wear the Bigdei Kehunah outside of the Azarah!
Hadar Zekenim #1 (19:18): The Isur to take them outside is only at the time of Avodah. This is somewhat like 'Kehunah applies to Aharon's descendants only when they wear the Bigdei Kehunah.'
Hadar Zekenim (19:18) #2: It was a Hora'as Sha'ah [to permit wearing them outside].