St Francis de Sales explains the reasons of Mary’s visit to her cousin Elizabeth

“Upon hearing from the same holy Gabriel that her kinswoman Elizabeth had in her old age conceived a son [Lk. 1:36], she desired, being her relative, to go to see her, so that she might wait upon her and be a comfort to her during her pregnancy, for she knew that such was the divine will…thus, aware of divine inspiration, she started out. She was not drawn by any curiosity to see if what the angel had told her was really true, for she had not the least doubt about it, but rather she was quite certain that things were exactly as he had declared.

It was these two virtues (charity and humility) which motivated her, and made her leave her little Nazareth, for charity is never idle; it burns in the hearts where it dwells and reigns, and the most blessed Virgin was full of it, because she bore Love Itself in her womb. She made continual acts of love, not only for God…she also loved her neighbor in a most perfect degree.

Our Lady went quickly that he might be sanctified, and that this sacred Child who was God, to whom alone belongs the sanctification of souls, might during this visit bring it about in the glorious St. John, purifying him and delivering him from Original Sin…It was charity, therefore, which made the Most Blessed Virgin cooperate in this sanctification. But it is no wonder that her sacred heart was quite full of love and desire for the salvation of people, since she bore within her chaste womb Love Itself, the Saviour and Redeemer of the world.

Along with charity, she was gifted with a profound humility, as it testified by those words which she uttered upon His servant in her lowliness, all ages to come will praise her and call her blessed…The most holy Virgin, hearing what her kinswoman Elizabeth said in her praise, humbled herself and referred all the glory to God. Then declaring her that all her happiness, as I said, came from the fact that He had looked upon His servant in her lowliness, she intoned that beautiful and wonderful canticle, the Magnificat [Lk. 1:46-55]”

Saint Francis de Sales (d.1622), Sermon for the Feast of the Visitation (excerpts).