If Jesus is just a prophet, how did he die on the cross and rise? Why does the Bible say that He said that the He was the Son of God?

Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

Jesus, son of Mary, is the slave of Allah and His word to Mary. He is not God, neither is he the son of God. There are many verses in the Quran proving this fact.
Allah says: { And there is none of the people of the Scripture (Jews and Christians), but must believe in him ['Iesa (Jesus), son of Maryam (Mary), as only a Messenger of Allâh and a human being],[] before his ['Iesa (Jesus) or a Jew's or a Christian's] death[] (at the time of the appearance of the angel of death).} [4:159]
When the Jews and the Roman's emporor plotted to kill Jesus. Allah made one of the people present similar in everything to Jesus. So, they killed the man who looked like Jesus. They did not kill Jesus. Jesus was raised alive to the Heavens. Allah says: {And because of their saying (in boast), "We killed Messiah 'Iesa (Jesus), son of Maryam (Mary), the Messenger of Allâh," - but they killed him not, nor crucified him, but the resemblance of 'Iesa (Jesus) was put over another man (and they killed that man), and those who differ therein are full of doubts. They have no (certain) knowledge, they follow nothing but conjecture. For surely; they killed him not [i.e. 'Iesa (Jesus), son of Maryam (Mary)} [4:157]
He will come back to earth before the Day of Judgement and will establish Islamic Sharia.
Allah says: {O people of the Scripture (Jews and Christians)! Do not exceed the limits in your religion, nor say of Allâh aught but the truth. The Messiah 'Iesa (Jesus), son of Maryam (Mary), was (no more than) a Messenger of Allâh and His Word, ("Be!" - and he was) which He bestowed on Maryam (Mary) and a spirit (Rûh)[] created by Him; so believe in Allâh and His Messengers. Say not: "Three (trinity)!" Cease! (it is) better for you. For Allâh is (the only) One Ilâh (God), Glory be to Him (Far Exalted is He) above having a son. To Him belongs all that is in the heavens and all that is in the earth. And Allâh is Allsufficient as a Disposer of affairs.} [4:171]
Al- Bukhari and Muslim reported that 'the Prophet (Sallallahu Alaihi wa Sallam) said: " Allah's Apostle said, "By Him in Whose Hands my soul is, son of Mary (Jesus) will shortly descend amongst you people (Muslims) as a just ruler and will break the cross and kill the pig and abolish the Jizya (a tax taken from the non-Muslims who are in the protection of the Muslim government). Then there will be abundance of money and nobody will accept charitable gifts."
He will stay for 40 years. Then Allah will make him die and this is his death. So, Jesus was not crucified nor was he killed. In fact, he did not die. He is still alive thanks to the power of Allah the Almighty.
If we imagine that Jesus was killed, then crucified, then raised up, there is no evidence in this proving that he was God or a son of God. This is clear to anyone who accepts that Allah, the Almighty, is able to do anything He wishes and that when He wants something, He says to it: {Be and it happens immediately} .
As for your asking about why the Bible says that he is the son of Allah, the answer is that the books of the people of scripture have been altered and modified. Things were added to the Bible and things were taken from it. Any claim that Jesus is a son of Allah is a fabrication because the authentic Injeel, later destroyed, which was first revealed to Jesus did not include any such lies.
Allah knows best