John Gill Commentary Deuteronomy 6:13

"Thou shalt fear Jehovah thy God; and him shalt thou serve, and shalt swear by his name." — Deuteronomy 6:13 (ASV)

You shall fear the Lord your God, and serve him
Serve him through fear; not through slavish fear, a fear of hell and damnation; but through filial fear, a reverential affection for that God that had brought them out of a state of bondage into great and glorious liberty, out of Egypt into Canaan's land, out of a place of misery into a land of plenty; and therefore should fear the Lord and his goodness, and from such a fear of him serve him, in every part of worship, public and private, enjoined; this passage Christ refers to (Matthew 4:10)

and shall swear by his name ;
when they made a covenant with any, or were called to bear a testimony for the decision of any controversy which could not be otherwise finished; or whenever they took an oath on any account, which should never be taken rashly or on any trivial account, and much less falsely; it should be taken not in the name of any idol, or of any other but the true and living God; the Targum of Jonathan is, "in the name of the Word of the Lord, in truth you shall swear."