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.