‘And let the priests also who come near to Yahweh, sanctify themselves, lest Yahweh break forth on them.'
This refers to Moses and Aaron. They are the priests who approach Yahweh at this time (Exodus 19:24). Moses especially (Exodus 17:15), and Aaron with him, are the ones who at this stage mainly act as priests on behalf of the people. There were secondary priests, heads of their fathers' houses, but they were not to be allowed to approach on this holy mountain (Exodus 19:24). Yahweh is reminding the people of the exalted position of Moses and Aaron.
But before they do so even they must sanctify themselves. For they cannot come without that. So once he has gone down the mountain and been with the people Moses, before returning, must again wash his clothes, to remove the earthiness of being with the people, as must Aaron. Then they may again approach Yahweh.
“Let the priests who approach Yahweh.” This is emphasising that Moses and Aaron, as those who are to approach Yahweh, have their privileged access as priests to the people. This is the reason that they come before Him, because they are mediators for the people. Yet in this case they alone of all the priests are to be allowed this access. This brings out the unique holiness of this situation. But because of this very fact they must re-sanctify themselves.
But behind the specific situation is a general situation. All priests who approach Yahweh at any time must sanctify themselves. Indeed the words may be a quotation of words already given to Moses and written down in the statutes previously laid down (Exodus 15:25). The simple approach confirms their primitive form. They will soon be replaced by a more complex situation.
“Lest Yahweh break forth on them.” They too will be destroyed if they do not obey Yahweh's requirements (compare Exodus 4:24) or if they seek to approach him covered in earthiness.