Beyond all possible doubt, the context is Christ as a High Priest. This is clear by a cursory review. Therefore, As we approach the High Priest's ministry we need to identify the prayers and supplications he performed -- then how they were answered.
The day of atonement brings the high point of this ministry into view. There were three beasts to be slaughtered by the High Priest:
“Aaron is to offer the bull for his own sin offering to make atonement for himself and his household. Then he is to take the two goats and present them before the Lord at the entrance to the tent of meeting. He is to cast lots for the two goats—one lot for the Lord and the other for the scapegoat. (Leviticus 16:6-8, NIV)
According to Hebrews, Christ was the sacrifices all combined and was the Priest 'offering himself'. In this offering in the temple there was formalistic payers uttered. Alfred Edersheim the Jewish historian actually has claimed to know the exact prayers that were uttered at the time of Christ. When the ritual arrived at the third beast, the scape goat, this was the prayer that was offered by the High Priest:
Laying both his hands on the head of this goat, the high-priest now confessed and pleaded: ‘Ah, Jehovah! they have committed iniquity; they have transgressed; they have sinned—Thy people, the house of Israel. Oh, then, Jehovah! cover over (atone for), I intreat Thee, upon their iniquities, their transgressions, and their sins, which they have wickedly committed, transgressed, and sinned before Thee—Thy people, the house of Israel. As it is written in the law of Moses, Thy servant, saying: “For on that day shall it be covered over (atoned) for you, to make you clean from all your sins before Jehovah ye shall be cleansed.” ’ (The Temple - Its Ministry and Services as they were at the time of Christ by Alfred Edersheim, P317)
For the first sacrifice, for himself and his family the exact same formula was used but 'they' is replaced with 'I'.
Now regardless of the accuracy of this historical prayer it indicates the type of prayer that naturally the high priest would pray. The high priest was officiating for the sins of the people, and naturally offered a sacrifice with prayers.
When we approach the priesthood of Jesus and the prayers he offered, we must look at his whole life and the suffering he endured by humbling himself into human flesh. Then more particularly, of course as he prayed in the garden approaching the time of actually offering his soul and body for the infinite penalty of the sins of the world, he offered prayers with a deep cry and groaning. Then even more in line with the time of the actual event of his ministry, on the cross:
My God, my God, why have you forsaken me? Why are you so far from saving me, from the words of my groaning? O my God, I cry by day, but you do not answer, and by night, but I find no rest. (Psalm 22:1-2, ESV)
This was Christ's prayers in the garden and on the cross where they reached their highest pitch, but as a high priest they were the prayers also of his whole ministry while in the days of his flesh on this earth.
I am poured out like water, and all my bones are out of joint; my heart is like wax; it is melted within my breast; my strength is dried up like a potsherd, and my tongue sticks to my jaws; you lay me in the dust of death. For dogs encompass me; a company of evildoers encircles me; they have pierced my hands and feet — I can count all my bones —they stare and gloat over me; (Psalms 22:14-17, ESV)
Having considered the context of his prayers as a high priest a shortsighted glimpse might ask, 'but these were prayers for himself, not for us?' Of course, absolutely not, they were prayers for the success of his ministry, for himself -- for us. Which naturally leads to the next verse:
Although he was a son, he learned obedience through what he suffered. And being made perfect, he became the source of eternal salvation to all who obey him, being designated by God a high priest after the order of Melchizedek. (Hebrews 5:8-10, ESV)
Now that we have understood the context, the answer to the question 'How his priestly prayers were heard' is obvious? His sacrifice was accepted and he became a priest forever. God was pleased with his work and rested in it, granting us access into his rest forever, as nothing more will ever be required to restore the fallen creation.
John Owen distinguishes between his payer which was hypothetical, 'Father if it be thy will' and 'absolute.' The prayer which was 'heard' was both hypothetical and absolute. For hypothetical, 'his mind was fortified against the dread and terror of nature'. More directly and importantly to the question Christ was heard absolutely:
The chief and principal supplications which he offered up to him who was able to save him from death were absolute; and in them he was absolutely heard and delivered. For upon the presentation of death unto him, as attended with the wrath and curse of God, he had deep and dreadful apprehensions of it; and how unable the human nature was to undergo it, and prevail against it, if not mightily supported and carried through by the power of God. In this condition it was part of his obedience, it was his duty, to pray that he might be delivered from the absolute prevalency of it, that he might not be cast in his trial, that he might not be confounded nor condemned. This he hoped, trusted, and believed; and therefore prayed absolutely for it, Isa. 50:7, 8. And herein he was heard absolutely. (Hebrews, John Owen, Vol 4, P509)