John 7:37-44 (ESVST)
37 On the last day of the feast, the great day, Jesus stood up and cried out, “ If anyone thirsts, let him come to me and drink.
On the last day of the feast. Remember that the feast that John is talking about here is the feast of booths. The word “booths” is a feminine singular noun meaning a booth, a thicket. This term is used for temporary shelters used to cover animals (Gen. 33:17); warriors (2 Sam. 11:11); and the prophet Jonah (Jon. 4:5). A specialized usage is employed for booths constructed for the fall harvest festival (Lev. 23:42, 43). The festival was known as the hag hassukkôt (H2282), the Feast of Booths (Deut. 16:13, 16). This was to remind the Israelites that they lived in booths when the Lord brought them up from Egypt (Lev. 23:43). This festival was held every on the 15th day of the 7th months for seven days and on the 8th day, they would hold a holy gathering and give food offerings to God.
So, on the last day here may refer to the 8th day. On this day, which John calls the great day, Jesus stood up and cried out. The word cried out means to scream or to speak loudly.
Jesus spoke loudly and said if anyone is thirsty, they must come to him and drink. Jesus made a similar statement in chapter 4
John 4:14 (ESVST)
but whoever drinks of the water that I will give him will never be thirsty again. The water that I will give him will become in him a spring of water welling up to eternal life.”
A tradition grew up in the few centuries before Jesus that on the seven days of the Feast of Booths, or Tabernacles, a golden container filled with water from the pool of Siloam was carried in procession by the high priest back to the temple. As the procession came to the Water Gate on the south side of the inner temple court, three trumpet blasts were made to mark the joy of the occasion and the people recited Isa. 12:3, “With joy you will draw water from the wells of salvation.” At the temple, while onlookers watched, the priests would march around the altar with the water container while the temple choir sang the Hallel (Ps. 113–118). The water was offered in sacrifice to God at the time of the morning sacrifice. The use of the water symbolized the blessing of adequate rainfall for crops. Jesus used this opportunity to call them to the real living water which was himself. If the rain gave life to the crops which would eventually be eaten by them and fade away, he himself was the living water not for the crops but for their souls. The water that he will give them will be an eternal spring welling up to eternal life. The words that Jesus used was from
Isaiah 55:1 (ESVST)
1 “ Come, everyone who thirsts, come to the waters; and he who has no money, come, buy and eat! Come, buy wine and milk without money and without price.
38 Whoever believes in me, as the Scripture has said, ‘Out of his heart will flow rivers of living water. ’” 39 Now this he said about the Spirit, whom those who believed in him were to receive, for as yet the Spirit had not been given, because Jesus was not yet glorified.
As the scripture said, simply referring to the old testament which did say that people should believe in him, Jesus is saying, those that believe in me will receive the Holy Spirit. Jesus was using the flowing of living water as the pouring of the Holy Spirit that will happen once he has died, resurrected and ascended to heaven.
40 When they heard these words, some of the people said, “ This really is the Prophet.” 41 Others said, “ This is the Christ.” But some said, “ Is the Christ to come from Galilee? 42 Has not the Scripture said that the Christ comes from the offspring of David, and comes from Bethlehem, the village where David was?” 43 So there was a division among the people over him. 44 Some of them wanted to arrest him, but no one laid hands on him.
By the prophet, the crowd was referring to the prophet like Moses who is mentioned in Deutronomy. Others said he was the Christ but had a problem with the fact that they thought that Jesus was born in Galillee and not in Bethlehem as the scripture said.
Micah 5:2 (ESVST)
But you, O Bethlehem Ephrathah, who are too little to be among the clans of Judah, from you shall come forth for me one who is to be ruler in Israel, whose coming forth is from of old, from ancient days.
This shows how ignorant these people were when it came to Jesus. What they did not know was that Jesus was indeed born in Bethlehem of Judea.