The people were upset NOT because, having read Isaiah's messianic prophesy, Jesus asserted that it was about Him and that He was the promised Messiah. On the contrary, as we read, they took this note from Him with acceptance and even enthusiasm.
The reason for their radical and abrupt change of minds was that what Jesus said afterwards was completely at odds with their expectation of what the Messiah should be up to: they expected a political-national Messiah who would have led a successful rebellion against the Romans (like Judas the Galilean, for instance in 6 AD), the very success itself of this rebellion to be the proof that He was really the God-chosen Messiah.
However, Jesus destroys all their expectations by asserting that He did not come exclusively to Jews, and not for sure to lead any revolution against Romans, and that's why brought an example of the Old Testament prophets, who were received by non-Jews, who had good hearts. By those examples He radically changes the semantics of "salvation": "I came to save representatives of all nations, to all people who will accept me with good hearts, not only those of the Jewish nation". But then also the representatives of Romans, to be sure, that is to say, of the conquerors and oppressors. Now, what does it mean that He came to bring "salvation" also to Romans who are not oppressed politically at all, being the most powerful Imperial power of the epoch? It can only mean that He came to save all mankind from a non-political oppression, from the oppression and infection of sin, of Satan, of the fallen nature and its self-destructive inclinations. Now, by these the entire humanity is oppressed and only Jesus can provide a rescue, a salvation.
But this universal messianship was not acceptable from the nationalistic expectations of Jews present in this synagogue. In fact, it was killing in them their most cherished dream of a delectation that would follow a just national vengeance against the Roman oppressors, for, in fact, what can be a greater enjoyment and delectation than to see your enemy prostrated before you, in front of the glorious Messiah-King, who would decisively beat them in a pitched battle and subdue them so as to make them lick the dust from His feet (Isaiah 49:23).
Now, Jesus says that, yes, He is the very Messiah promised by Isaiah, but not with such a crude exegesis of Isaiah's prophesy, for it is not at all in His agenda to fulfil the political-national expectations of Jews.
Of course, the Jews present in this synagogue got furious at the prospect of seeing this impostor debunking their most cherished religious-political-national expectations, for the latter they thought to be blessed by God Himself. Jesus, thus, appeared in their eyes as a traitor of the highest scale, that of God and God's chosen people.
Yet, since they had before that heard about Him working miracles and since they were bewildered by the grace of His words, in deep recesses of their hearts they could suspect at least that He indeed was whom He said to be, but it was so painful for them to bear such a disappointment, that they would rather kill such Messiah, even if He was the real one, rebel against God who sent Him, but not to accept the abolition of the prospect of the delectation of the vengeance over Romans and the subsequent grandeur of the restored Jewish Kingdom under the Messianic King.
Jesus did not destroy, in fact, the prospect of delectation but gave to it a different, more universal and glorious dimension: all humans can participate in the delectation and joy of defeat of the kingdom of sin and Satan through Him, the Heavenly King, and become co-Kings with Him in this eternal Kingdom with victors from all nations of the world.
Something similar happened in Jonah the prophet, who rebelled against God and ran away from His order to preach to the Ninevians, the enemies of Jews; even, Jonah was ready to kill himself but not to do such an unpleasant thing. Jesus was similarly unpleasant for the Jews present in this synagogue. No surprise, in this perspective, to see them attempting to murder Him.
P.S. One philosopher said that Jesus was crucified by Jews before His actual crucifixion by their wrong and petty idea of Messiah: they saw Him not as a Heavenly King, but as an earthly king. Even His own disciples could not fully understand Him, thus crucifying Him by their wrong ideas about Him (cf. Matthew 20:21).