"And when He had come into Jerusalem, all the city was moved, saying, 'Who is this?' So the multitudes said, 'This is Jesus, the prophet from Nazareth of Galilee.' Then Jesus went into the temple of God and drove out all those who bought and sold in the temple, and overturned the tables of the money changers and the seats of those who sold doves. And He said to them, 'It is written, "My house shall be called a house of prayer," but you have made it a "den of thieves."' Then the blind and the lame came to Him in the temple, and He healed them. But when the chief priests and scribes saw the wonderful things that He did, and the children crying out in the temple and saying, 'Hosanna to the Son of David!' they were indignant and said to Him, 'Do You hear what these are saying?' And Jesus said to them, 'Yes. Have you never read, "Out of the mouth of babes and nursing infants You have perfected praise"?' Then He left them and went out of the city to Bethany, and He lodged there."
"Who is this?" - That was the question being asked by the rest of Jerusalem as Jesus entered in. Certainly the actions of such an excited crowd would be heard all over the city, and many would be wondering what was going on.
"This is Jesus, the prophet from Nazareth of Galilee" - Certainly that was an accurate statement, and some may have been referring to Him as the prophet promised by Moses (cf. Deut. 18:15-18). Of course, Jesus was much more than merely a prophet from Nazareth. He should have been proclaimed as the Messiah to those who inquired. How often do we today speak part of the truth, but, out of cowardice, fail to proclaim it to its fullest extent?
Jesus entered into the temple the day He triumphantly entered Jerusalem, but only to look around and make observations. He did not cleanse the temple until the next day (cf. Mark 11:11-17). Matthew evidently saw fit to include that information here (perhaps because of the subject matter), though it was slightly out of chronological order.
It is unknown exactly what day these miracles of healing were performed on. It is likely that He performed signs of this sort each day He was in the temple. These miracles certainly support the authority that He manifested in the temple through the cleansing of it (cf. Matt. 21:12). It should be emphasized that Jesus is no longer performing His miracles in the obscure towns and deserted places of Galilee. He is working His power to heal in the court of the temple while surrounded by His most bitter enemies.
The chief priests and scribes were very angry when they saw the wonderful things that Jesus did and the praise the children were giving Him. They ask Jesus if He is hearing what the people are saying about Him. The implication is that they believe it to be inappropriate and that Jesus should do something to stop the children from making these statements (cf. Luke 19:39,40).
Jesus defends the statements of the children by quoting part of Psalm 8:2 from the Septuagint. Jesus' presence in the temple should have been announced with joy by all, not just by the "babes and nursing infants" (cf. Matt. 18:3,4).
After inspecting His Father's house (i.e., the temple; cf. John 2:16), He traveled back to Bethany to spend the night there. Certainly His friends that resided there would have desired His company, and it would not have been safe for Him to spend the night in Jerusalem (due to the animosity of the religious leaders).