Why did Virgin Mary call herself the servant of the Lord?

She calls herself a "handmaid" (ancilla, δούλη) in Luke 1:38 and in her Magnificat canticle (Luke 1:48) because she is very humble.

Fr. Cornelius à Lapide, S.J., commentates Luke 1:38:

Mark the humility, modesty, and resignation of the Virgin, for though saluted by the angel as Mother of God, she calls herself His handmaid, not His mother; handmaid by nature, mother by grace. Pet. Dam. (Serm. 3 de Nativ. Virg.) And S. Bernard (Serm. in Apoc. 12) says, “A great sign: deservedly is she made mistress of all who declared herself servant of all.”

and

Ver. 48.—For He hath regarded, &c. S. Augustine (super Magnificat) says, “This is the grace of her exultation, that He hath regarded the low estate of His handmaiden: it is as if she said, because I exult in His grace, therefore my exultation is from Him; and because I love His gifts on account of Himself, therefore I exult in Him. S. Bernard (Serm. 57 in Cant.) says, “God regards the earth and causes it to tremble; He regards Mary and infuses grace. He hath regarded, she says, the lowliness of His handmaiden, for, behold, from henceforth all generations shall call me blessed. These are not the words of one lamenting or fearing, but of one rejoicing. Hence He says to her, Arise quickly, my love, my dove, my fair one, and come away.”

Lowliness, or low estate. Vulgate, humilitatem; Greek, ταπείνωσιν. Humility here properly means lowliness of estate, not the virtue of humility as opposed to pride, for this is called ταπεινοφροσύνη; for humility alone among virtues is ignorant of itself; and he who boasts of his humility is proud, not humble.

Secondly, however, by humility may be understood the virtue itself of humility; for on account of this God had regard to the Blessed Virgin, and chose her for His mother; for a humble person recognises his virtues as being the gifts of God; wherefore among them he sees also his own humility, but he ascribes it not to his own strength, but to the grace which he had received from God.

As, therefore, the Blessed Virgin here recognises her election to be the Mother of God (which was a far greater thing), so likewise she recognises that she was fittingly adorned for so great a dignity by her humility, virginity, and other virtues which had been imparted to her by God. For a humble person recognises his own low estate, his misery, his poverty, yea, even his own nothingness, and ascribes all that he is and has to God, Whose he is, and says with the Psalmist, Not unto us, O Lord, not unto us, but unto Thy Name give the glory.

Listen to S. Augustine (Serm. 2 de Assump.), “O true humility which hath borne God to men, hath given life to mortals, made new heavens, and a pure earth, and given liberty to the souls of men. The humility of Mary was made the heavenly ladder by which God came down to earth. For what does regarded mean but approved? For many seem in the sight of men to be humble, but their humility is not regarded by the Lord. For if they were truly humble, then they would not wish to be praised by men, and their spirit would not rejoice in the world but in God.” And S. Chrysostom (Hom. 2 in Ps. 50) says, “The greatest sacrifice or all is humility, for the same man who by sinning has separated himself from God, subjects himself to Him by humility, when he is converted to penitence.” And lastly S. Bernard says, “It is humility which truth begets for us, and it has not heat, and it is humility which love forms and inflames. The latter consists in affection, the former in knowledge: by the former we learn that we are nothing, and we learn it from ourselves and our own weakness; by the latter we tread underfoot the glory of the world, and we learn it from Him Who emptied Himself, and Who, when men sought to make Him a king, fled; but when He was sought for reproaches and for the Cross, He did not flee, but offered Himself willingly.” The Blessed Virgin had both these (humility and love) in an eminent and heroic degree.