The Scriptures do NOT record that the disciples HEARD Jesus’ prayers. The Gospel writers/narrators tell US what the words were, but again the Gospel writers/narrators do NOT state that the disciples HEARD those words. So we are left to speculate how the Gospel writers/narrators came to know what those words were. It is, of course, POSSIBLE that the disciples DID hear the words, but it is also possible they didn’t. We can look at each in turn.
Assuming it is possible for the disciples to hear the words, how? The article you linked to heads us down the right road on the “how,” even though it is dead wrong on the assertion that Scripture definitely tells us that they heard. Jesus may well have been praying quite loudly. Furthermore, the words recorded of Jesus’ prayers are surely not ALL that He prayed. It seems highly unlikely that Jesus would walk a stones’ throw away; pray one sentence; walk back a stones’ throw, upbraid Peter, James, and John; walk a stone’s throw away again; pray the same one sentence; walk back a stones’ throw yet again; and upbraid Peter, James, and John again. Rather, He probably prayed quite extensively each time. The disciples heard part of those prayers before they fell asleep. Nor is it likely that the words recorded are even all that they HEARD, as if they managed to stay awake for exactly one sentence each time. Instead, these are the only words that they chose to (or I would say, “were inspired to”) record.
Now let’s turn to the other option: the disciples did not hear Jesus’ words. How did the Gospel writers/narrators come to learn of them? Again, I think there are two possibilities.
First, they may have received direct revelation from the Holy Spirit. After all, it has been the claim of the Church for 2,000 years that the Bible is inspired by God. In fact, there are MANY stories in Scripture about which we must speculate as to how the writer acquired his data, but some of those situations require quite strained speculations or speculations in which information is passed on from 3 or 4 mouths before it could have made it to the writer. Direct revelation may well be the more likely answer in these narratives.
Second, Jesus Himself could have told them. Remember, He walked the earth 40 days between His resurrection and ascension. Remember three incidents especially.
The first incident we should remember is the scene between Jesus and Peter on the beach when Jesus forgives Peter for his failure in betraying Him. Is it not also possible and perhaps even likely that sometime during the 40 days Jesus also explicitly forgave Peter, James, and John for their failure in the garden? If so, He might easily have recounted the prayers they had not heard (something like, “Even when I was praying for the cup to pass, you could not stay awake.”).
The second incident occurred when He walked with 2 of His disciples on the road to Emmaus, and He “expounded to them in all the Scriptures the things concerning Himself”—that is, He expounded how He fulfilled various things in the Law and the Prophets. The prayer that Jesus prayed (“Take this cup away from Me: nevertheless, not what I will, but what You will”) is seen as a fulfillment of Isa. 50:5-6 (“The Lord GOD has opened My ear; And I was not rebellious, Nor did I turn away. I gave My back to those who struck Me, And My cheeks to those who plucked out the beard; I did not hide My face from shame and spitting.”) (In fact, going back to the idea that perhaps the disciples COULD hear and DID hear part of Jesus’ prayer before they fell asleep, this would explain why it was only this part they recorded—it fulfilled prophecy.)
The third incident was similar to the second. When the Emmaus disciples related their encounter with the risen Lord to other disciples, Jesus appeared in their midst and again explained how His life had fulfilled various prophesies.
And there may be other ways the Gospel writers/narrators could have learned of the content of Jesus’ prayers that haven’t occurred to me.
"When the Round Table is broken every man must follow Galahad or Mordred; middle
things are gone." C.S. Lewis
“The chief purpose of life … is to increase according to our capacity our knowledge of God by all the means we have, and to be moved by it to praise and thanks. To do as we say in the Gloria in Excelsis ... We praise you, we call you holy, we worship you, we proclaim your glory, we thank you for the greatness of your splendor.” J.R.R. Tolkien