According to Catholicism, what does 'come upon thee' and 'overshadow thee' mean?

To supplement the answer above, I will provide more Catholic insight to the overshadowing and its significance to Catholic theology and especially the Annunciation.

From the very beginning, the Lord has used Spirit and Water when starting a New Creation. From the very moment that our God created the universe you can see the overshadowing of the Holy Spirit.

Genesis 1:1-2 In the beginning God created heaven, and earth. And the earth was void and empty, and darkness was upon the face of the deep; and the spirit of God moved over the waters.

In the same way, Mary’s womb, being void and empty of the seed of men, was filled with the Holy Spirit.

The most significant foreshadowing of the Annunciation, Mary, and New Covenant Christianity is that of the Ark of the Covenant and the descriptions of what it held and how the Spirit of God would dwell with it overshadowing it.

Exodus 40:31-35 He set up also the court round about the tabernacle and the altar, drawing the hanging in the entry thereof. After all things were perfected, the cloud covered the tabernacle of the testimony, and the glory of the Lord filled it. Neither could Moses go into the tabernacle of the covenant, the cloud covering all things and the majesty of the Lord shining, for the cloud had covered all. If at any time the cloud removed from the tabernacle, the children of Israel went forward by their troops: If it hung over, they remained in the same place.

Significance of what is overshadowed:

Old -- New
The supernatural bread, the Manna -- The True Bread of Life
The staff of Aaron, the symbol of the High Priest -- The True High Priest
The tablets which held the very words of God -- The True WORD of God

The promises of Gabriel to Mary are the New Covenant version of the promises to Noah and the Flood, to Abraham concerning the birth of his son Isaac in the barren womb of Sarah, and to Moses that God would lead his people from Slavery overshadowed and through water.

The honor given to Mary is foreshadowed but exceedingly greater than that given to the Ark of the Old Covenant. Where God forbade that sinful man touches the Ark of the Old and they did and were punished, sinful man was prevented from ever touching Mary who is the Ark of the New Covenant. Greater than the bowls and plates and censers sanctified for the use of former sacrifices and offerings, Mary as the fulfillment of all previous things is ever more set apart for the Purposes of God.

Did the Lord continuously overshadow the Ark Of Old, or did he dwell there continuously? He did indeed dwelt there continuously, preventing people to even look upon it, so much so that the High Priest would tie a rope around himself so that if he was struck down he could be pulled from the Holy of Holies.

The Lord also overshadowed Mary throughout her life, protecting her for all time for his purpose from sin, and set her apart to be the woman whose heel would crush the head of the serpent, the Mother of the Second Person of the Trinity, Mother of the Twelve Tribes Reunited, and Mother of us all. Mary continues to represent the Church, the Body of Christ, and continues to be overshadowed, set apart, and cherished by her Son.

The phrases you point out indicate to the Catholic Church that the Holy Spirit was responsible for Jesus' conception:

The Annunciation to Mary inaugurates "the fullness of time," the time of the fulfillment of God’s promises and preparations. Mary was invited to conceive him in whom the "whole fullness of deity" would dwell "bodily." The divine response to her question, "How can this be, since I know not man?" was given by the power of the Spirit: "The Holy Spirit will come upon you."

(Catechism of the Catholic Church, paragraph 484)

Or again:

The Spirit comes upon the Virgin Mary and "overshadows" her, so that she might conceive and give birth to Jesus.

(Catechism, paragraph 697)

Aquinas, in his Catena Aurea, quotes Gregory of Nyssa on this passage:

O blessed is that womb which because of the overflowing purity of the Virgin Mary has drawn to itself the gift of life! For in others scarcely indeed shall a pure soul obtain the presence of the Holy Spirit, but in her the flesh is made the receptacle of the Spirit. ... Moreover, the power of the Highest shall overshadow you. Christ is the power of the most high King, who by the coming of the Holy Spirit is formed in the Virgin.

(emphasis added)

One sees the same thing in the Creeds:

For us men and for our salvation
he came down from heaven,
and by the Holy Spirit was incarnate of the Virgin Mary

(Nicene Creed)

and in Jesus Christ, his only Son, our Lord,
who was conceived by the Holy Spirit

(Apostles' Creed)

(emphasis added)