Deuteronomy 10:8

At that time the Lord distinguished the tribe of Levi, because they had been zealous (even) to slay for His honour; that they should bear the ark of the Lord’s covenant, and stand before the Lord to minister unto Him and to bless in His Name until this day.

The Levites were to minister before Him together with the descendants of Aaron.

to bless in His name for Eleazar used to give the Priestly Blessing [cf. Numbers 6: 24 ff.].

AT THAT TIME THE ETERNAL SEPARATED THE TRIBE OF LEVI. “This refers back to the first subject. At that time, in the first year of your departure from Egypt when you sinned by worshipping the calf and the sons of Levi did not sin, G-d separated them from you.” This is Rashi’s language. And if so, he is stating that from that time the sons of Levi were separated before Him, blessed be He [but only in thought], for He did not separate them in fact until the second year in the second month [after the first census which took place on the first day of the second month, in the second year after they were come out of the land of Egypt]! Or it may be that he is alluding to what he said to them [at the affair of the calf], Consecrate yourselves today to the Eternal, for thereby you dedicate yourselves to the service of priests and Levites.
The correct interpretation, it appears to me, is that the expression at that time means that the second [giving of the Tablets] was like the first. For, from the time He said to Moses Hew thee two Tablets of stone,, 34:1. He was reconciled with Israel, and Moses was instructed regarding the construction of the Tabernacle. Then He was reconciled also with Aaron, and Moses was commanded, and they shall make holy garments for Aaron thy brother, and his sons — to sanctify him — that he may minister unto Me in the priest’s office,, 28:4. The phrase to sanctify him is found there in Verse 3. and He further commanded him concerning the initiation [of the priests]. Therefore he said [here in Verse 6] that Eleazar ministered in the priest’s office in his father’s stead, since their anointing was to them for an everlasting priesthood throughout their generations., 40:15. Moses regarded the separation of the entire tribe [of Levi — priests and Levites] as one matter [therefore he said here, At that time — when the second Tablets were given, even though only the priests were formally separated, it was equivalent to a designation of the entire tribe, with the formal designation of the Levites to come after the first census], for, from that day on it was [resolved] before Him that His tribe be wholly given unto Him to perform the service of the Tabernacle. Or perhaps he is saying that the tribe of Levi was separated at that time for the priests to be [chosen] from them, for they [i.e., the priests] also carried the ark. … Now, according to the interpretation …

בעת ההיא הבדיל ה' את שבט הלוי, the time Moses refers to is that of the sin of the golden calf when the latter had found grace in the eyes of the Lord. (compare Exodus 32,26)

ולברך בשמו this refers only to the priests who are to bless the rest of the Israelites.

בעת ההיא הבדיל, after G’d had already set aside the tribe of Levi as the tribe to perform the holy service in the Temple so that you are not deemed worthy for this task, my prayer was not effective enough except to prevent G’d from destroying you, so that He said to me: “arise and make them journey, and be in front of the people!” (verse 11) The expression is used in the same sense as in Numbers 10,2)

בעת ההיא, at that time, i.e. when the golden calf had been made, as I explained earlier.

הבדיל ה' את שבט הלוי, “the Lord separated the tribe of the Levites,” in order for it to serve the Lord specifically. The reason was that that entire tribe had not participated in the sin of the golden calf. This is what the prophet Ezekiel referred to in Ezekiel 44,15: והכנים הלויים בני צדוק אשר שמרו את משמרת מקדשי בתעות בני ישראל מעלי, המה יקרבו אלי לשרתני, “but the Levite-kohanim- descendants of Tzadok, who safeguarded the charge of My Sanctuary when the Children of Israel strayed from Me, etc., they will draw near to Me to serve Me.” As a result of the Levites having been appointed to become G–d’s share and inheritance on earth, they did not receive ancestral land in the Holy Land, [they would not have had time to work it, anyway Ed.] If you will therefore ask how they would support themselves economically, the answer is that G–d commanded the landowners to give 10% of their crops to the Levites and they were allocated a number of cities in the land of Israel for their exclusive use, houses with gardens attached for growing vegetable. (48 such cities)