Why did Jesus need to die for our sins? I feel silly asking this because I feel like it should be obvious. Why did Jesus need to die for our sins?
I am asking because this part of Christianity still confuses me. I thought Judaism taught that we will go to heaven if we believe in Judaism, but after a purification (by fire??) of our earthly sins. So Jesus' dying also guaranteed that we would go to heaven, but weren't believers guaranteed entry into heaven to begin with?
So does that mean that Jesus sacrificed himself on the cross so we would not have to go through the purification process? Is this the reason he died on the cross? Otherwise we all could have just become Jews and went through the purification process to go to heaven. Please help me understand what I am missing because I feel this can't be right.
A Baptist would point back to the garden of Eden and the nature of the problem.  There are sins that a person commits by an act of the will in opposition to what is known to be right and there are sins committed in some degree of ignorance, both of omission and commission.  Remedies for these (more accurately atonement of these) were prescribed for God's people, the Jews, in the levitical law and the sacrificial system.
Requisite was the offering of a sacrifice according to the penitent's means and the nature of the offense.  A reading of Leviticus is useful here as a manual regrading the levitical system. Genuine repentance for an act on the part of the transgressor is an integral part of the offering and is sometimes explicit and sometimes implicit but always required.  Often God expresses displeasure over the offerings being given for the expressed reason that &quot;their hearts are far from me&quot;.
This brings us to what lies beneath.  When Adam fell he did more than disobey God, he internalized rebellion.  His created nature changed when he decided to take for himself that which is the purview of God alone:  knowledge of good and evil.  Since like begets like, all of Adam's progeny (humanity) are born with this rebellion within the fabric of their natures.  Oswald Chambers, a Scottish Baptist (1874 - 1917) describes this sin nature as a disposition:

The Bible does not say that God punished the human race for one man’s sin; but that the disposition of sin, viz., my claim to my right to myself, entered into the human race by one man, and that another Man took on Him the sin of the human race and put it away (Hebrews 9:26) — an infinitely profounder revelation. The disposition of sin is not immorality and wrong-doing, but the disposition of self-realization — I am my own god. This disposition may work out in decorous morality or in indecorous immorality, but it has the one basis, my claim to my right to myself. - The Bias of Degeneration

It is for the destruction of the power of this disposition in humanity that Christ died.  He did not come only to expiate each and every individual sin (which we all need) but to break the power of that which binds us all, to call us all to turn to the Living God through himself, to give us the gift of the Holy Spirit and thereby, to set us free from the law of sin and death.

For the law of the Spirit of life has set you free in Christ Jesus from the law of sin and death. For God has done what the law, weakened by the flesh, could not do. By sending his own Son in the likeness of sinful flesh and for sin, he condemned sin in the flesh, in order that the righteous requirement of the law might be fulfilled in us, who walk not according to the flesh but according to the Spirit. For those who live according to the flesh set their minds on the things of the flesh, but those who live according to the Spirit set their minds on the things of the Spirit. For to set the mind on the flesh is death, but to set the mind on the Spirit is life and peace. 
For the mind that is set on the flesh is hostile to God, for it does not submit to God's law; indeed, it cannot. Those who are in the flesh cannot please God. You, however, are not in the flesh but in the Spirit, if in fact the Spirit of God dwells in you. Anyone who does not have the Spirit of Christ does not belong to him. But if Christ is in you, although the body is dead because of sin, the Spirit is life because of righteousness. If the Spirit of him who raised Jesus from the dead dwells in you, he who raised Christ Jesus from the dead will also give life to your mortal bodies through his Spirit who dwells in you. - Romans 8:2-11

Jesus is the once for all sacrifice that all of the law and the prophets foreshadowed and pointed towards.  Under the Law one must offer sacrifice continually, day by day and year by year in order to redress transgression and sin because it dealt on the order of action.  Jesus sacrifice was on the order of disposition to reconcile us once for all.

But in these sacrifices there is a reminder of sins every year. For it is impossible for the blood of bulls and goats to take away sins. Consequently, when Christ came into the world, he said, “Sacrifices and offerings you have not desired, but a body have you prepared for me; in burnt offerings and sin offerings you have taken no pleasure. Then I said, ‘Behold, I have come to do your will, O God, as it is written of me in the scroll of the book.’” When he said above, “You have neither desired nor taken pleasure in sacrifices and offerings and burnt offerings and sin offerings” (these are offered according to the law), then he added, “Behold, I have come to do your will.” He does away with the first in order to establish the second. And by that will we have been sanctified through the offering of the body of Jesus Christ once for all. And every priest stands daily at his service, offering repeatedly the same sacrifices, which can never take away sins. But when Christ had offered for all time a single sacrifice for sins, he sat down at the right hand of God, waiting from that time until his enemies should be made a footstool for his feet. For by a single offering he has perfected for all time those who are being sanctified. - Hebrews 10:3-13

In the temple there was a thick curtain separating the holy place (ritual service) from the most holy place (where God's presence dwelt):

And you shall make a veil of blue and purple and scarlet yarns and fine twined linen. It shall be made with cherubim skillfully worked into it. And you shall hang it on four pillars of acacia overlaid with gold, with hooks of gold, on four bases of silver. And you shall hang the veil from the clasps, and bring the ark of the testimony in there within the veil. And the veil shall separate for you the Holy Place from the Most Holy. You shall put the mercy seat on the ark of the testimony in the Most Holy Place. - Exodus 26:31-34

Even in the Levitical system that God instituted there was no personal access to God Himself except through religious ritual and faith that God would express mercy through heartfelt obedience.  In Matthew's Gospel account of the death of Jesus Christ it is recorded that, at the moment of Jesus' death, the veil in the temple was torn from top to bottom:

And Jesus cried out again with a loud voice and yielded up his spirit. And behold, the curtain of the temple was torn in two, from top to bottom. And the earth shook, and the rocks were split. - Matthew 27:50-51

The Epistle to the Hebrews declares that Jesus condemned and removed the sin within humanity that separates us from God which the veil typified:

Therefore, brothers, since we have confidence to enter the holy places by the blood of Jesus, by the new and living way that he opened for us through the curtain, that is, through his flesh, and since we have a great priest over the house of God, let us draw near with a true heart in full assurance of faith, with our hearts sprinkled clean from an evil conscience and our bodies washed with pure water. - Hebrews 10:19-22

Sin causes separation between God and man and it is woven into the fabric of our flesh.  When the Son of God emptied Himself and veiled his glory within flesh he identified Himself with all humanity.  When he offered His own life to God as a spotless sacrifice (think Day of Atonement, Leviticus 16) He permanently removed the barrier restricting mankind's access to Almighty God.
Now that a better and permanent access has been opened there is no benefit for for those in Judaism to remain in the foreshadowing of religious ritual, serving in the Holy Place as though the veil of separation from God's presence still remained.
The following is from a Messianic Jewish blog entitled &quot;The WORD in Hebrew&quot;:

When the phrase “Completed Jew” is used, it’s used in the sense of “completed faith” – Unlike in other religions or heritage, when a Jew comes to faith in the Jewish Messiah, it is not a “conversion”, but a COMPLETION OF FAITH . Since Jesus was Jewish and came to the Jewish people to be their Messiah, There is nothing more Jewish a Jewish person can do than to follow Him. Believing and following the Jewish Messiah, YESHUA, is a completion of faith for the Jew.