What is the Catholic explanation of the particular temptations of Jesus in the wilderness? Please refer to Luke 4:1-13: Jesus fasts and is tempted in the wilderness

If Son, stone to bread > Kingdoms for you, if you pay me [Satan] homage > If Son, throw yourself, God will protect you.1

From Catholic theology, what is the explanation of the devil tempting Jesus with these temptations, and presenting them to him in a particular graduation?

1. Actual chronology may be as in Matt 4:1-11.

Related: Why Mathew and Luke Presented a Different Order of Satan's Temptation.



Just to clarify, this question is not seeking the Catholic explanation of why Jesus was tempted but why he was tempted with these temptations and in a certain order.

Explaining further, the devil might tempt one man with a certain sin - in the case of David it was adultery - and another man with a different one. The [Catholic] explanation being sought is why these particular temptations were presented to Jesus and in a particular order.
I imagine Catholics and Protestants are pretty much united in the way they look at Jesus' temptation in the wilderness, since both "camps" hold Scripture in high esteem and believe it to be God's word. 

There is perhaps no better answer to your question, then, than the one found in Hebrews, especially chapter 2, verses 17 and 18, as well as chapters 4 and 5, beginning with the section from 4:14 ("Therefore, . . .") to 5:10, which ends with Jesus being referred to as a "high priest according to the order of Melchizedek" (5:10). 

Of seminal importance are 2:17-18 and 4:15, which say:


  "Wherefore, it behoved [Jesus] in all things to be made like unto his brethren, that he might become a merciful and faithful high priest before God, that he might be a propitiation for the sins of the people. For in that wherein he himself hath suffered and been tempted he is able to succour them also that are tempted" (Douay-Rheims). 


And, 


  "For we have not a high priest who cannot have compassion on our infirmities: but one tempted in all things like as we are, without sin" (Douay-Rheims). 


Just as the ancient Hebrews were tested (and tempted) in the wilderness for 40 years, so too was Jesus tested (and tempted) in the wilderness for 40 days. There is significance in the two 40s, but that is not the point of the passage. The point of the passage is simply this:


  The reason Jesus can be an empathic high priest to His children is because He experienced every temptation we experience, yet He did so without sinning. 


In other words, no one can say Jesus' time on earth was a walk in the park. He, like us, learned obedience to the Father's will through suffering. Unlike us, however, He never stumbled in His obedience and never once gave in to the devil's blandishments and half-truths, whether in His wilderness experience or at any point in His earthly ministry. 

Despite Jesus' perfection, He knew firsthand what it is to be tempted. True, there was nothing in His character which was attracted by, or drawn to, sin. Nevertheless, His temptations were real, raw, and most significantly, perhaps, redemptive for us, particularly in the Garden of Gethsemane. It was there in His prayer to the Father that He expressed His desire to be exempted from the bitter cup He was about to drain on our behalf, yet He concluded His prayer with these words:


  "My Father, if this chalice may not pass away, but I must drink it, thy will be done" (Luke 22:42 Douay-Rheims).


Jesus, having emerged successfully from being tempted by the devil, despite the harshness of the wilderness, and despite being weakened by His 40-day fast, became an empathetic and compassionate high priest to all those who 


  ". . . go . . . with confidence to the throne of grace . . . [to] obtain mercy and find grace in seasonable aid" (Hebrews 4:16 Douay-Rheims).


Moreover, Jesus emerged from His temptation steeled by having defeated His arch enemy, and in so doing was prepared for His greatest temptation in the Garden of Gethsemane (i.e., "olive press"), where the pressure became so great that "his sweat became as drops of blood, trickling down upon the ground (Luke 22:44 Douay-Rheims).  

As for the significance of the temptations themselves and the "graduation" of them, as you put it, the three temptations seem to fit the pattern laid out for us by the disciple Jesus loved (viz., John) in 1 John 2:16-17 (NASB, Updated):


  "For all that is in the world [i.e., the kosmos], the lust of the flesh, and the lust of the eyes, and the boastful pride of life, is not from the Father, but is from the world. The world is passing away, and also its lusts; but the one who does the will of God lives forever."



The lust of the flesh corresponds to the temptation for Jesus to turn stones to bread.
The lust of the eyes corresponds to the temptation for Jesus to worship the devil in exchange for all the kingdoms of the world. (Notice that the devil showed Jesus these kingdoms in a moment of time, obviously not long enough for Jesus to see the corruption that existed in all those kingdoms.)
The boastful pride of life corresponds to the temptation for Jesus to cast himself from the pinnacle of the temple in Jerusalem.


In each temptation, the devil wanted Jesus to be and to act autonomously. Jesus chose instead to be obedient to and dependent on His Father. To Jesus, his food was to do only the will of his Father, and to complete the work His Father had laid out for him (see John 4:34). 

From eternity past Jesus had determined to humble himself. As Paul tells us in Philippians chapter 2:


  ". . . although He existed in the form of God, [He] did not regard equality with God a thing to be grasped, but emptied Himself, taking the form of a bond-servant, and being made in the likeness of men. Being found in appearance as a man, He humbled Himself by becoming obedient to the point of death, even death on a cross" (vv.6-8 NASB, Updated). 


In conclusion, while Jesus never ceased being God the Son while he was on earth, He did veil, if you will, the glory He possessed from all eternity by virtue of being coequal with the Father and the Holy Spirit. At some point in the future, however, during the Day of the Lord, Jesus will be reveal Himself to be the King of kings, and the Lord of lords, 


  ". . . so that at the name of Jesus EVERY KNEE WILL BOW, of those who are in heaven and on earth and under the earth, and that every tongue will confess that Jesus Christ is Lord, to the glory of God the Father" (Philippians 2:10-11 NASB Updated).