Nobody has given a proper "Biblical Studies" hermeneutic answer to this yet, so I will attempt to do so. This is an incredibly loaded question; however, that just gives me more space to educate. I will attempt to do so.
To understand why it had to be a death specifically, we must understand the religious context from which Jesus arose. We call this period the "Second Temple Period" because it was the time of the 2nd Jewish Temple. During this time period, there were many different, let's call them 'flavors', of what we will call "Second Temple Judaism".
It must be understood that Second Temple Judaism is distinctly different from what today we call Judaism. In contemporary speech, when a person says Judaism, they actually mean Rabbinic Judaism, a form of Judaism that developed in response to the destruction of the Second Temple as well as the rise of the Christians in a process we call "The Parting of the Ways", where the two groups diverged on how to deal with sin without a physical Temple, eventually becoming a diaspora nation with no homeland.
The main flavors of Second Temple Judaism were the Pharisees, the Sadducees, and the Essenes. There were more than this, however, but these three represent poles around which most other groups revolved. Each of these groups considered the first 5 books of the Old Testament to be sacred; however, not all of them agreed where the temple was supposed to be (specifically, the Sarmatians did not agree with the Temple being in Jerusalem). Despite these geographical disagreements, the function of the Temple remained the singular focus for all of them: it was the only place on earth where the gap between God’s holiness and human corruption could be bridged.
Now, all of these flavors of 2nd Temple Judaism had different cultures, came from different ranks of society, and had wildly different interpretations of scripture. But while they disagreed on the 'where' and 'how,' they all agreed on the problem: The Curse of Cain.
The Second Temple Jews understood that humanity was under a curse of violence begetting violence. They interpreted the story of Cain and Abel as the introduction of a curse that required blood to set right. In the ancient mind, life is in the blood, and when you take a life (sin), you create a "blood debt." The only way to pay a debt of life is with life. The sacrifice of the yearly Paschal Lamb at the temple was a temporary stop-gap on this accumulation of sin. The death served as a scrubbing agent to clean the pollution of sin that would otherwise consume the community; this is the Levitical concept of kippur (atonement) which means "covering".
As such, they had different expectations of who the Messiah was going to be and what role he was going to fulfill. The person to break the curse was to be the Messiah, the Savior. For the early Christians, Jesus was the answer to this ancient logic; he was the final Paschal Lamb, the only life with enough value to settle that debt of history. Jesus broke the cycle of debt from the inside.
In the early Jesus Movement (this is the term used for the followers of Jesus before the destruction of the 2nd Temple), all of these Messianic expectations came into play regarding how people viewed Jesus, during his lifetime, after his lifetime, and notably, after the destruction of the Second Temple. The latter part is notable because the Jesus Movement really took off after the destruction of the Second Temple. Had that event not happened, the followers of Jesus would not have had such a massive event to galvanize them.
It wasn't just the simple fact of its destruction, though, but what the Temple represented, how it related to Second Temple Jewish customs, and how the followers of Jesus were able to use it as proof of Jesus's foresight and the fulfillment of his prophecies. This event was viewed as the "Apocalyptic proof" that the mechanism of reality had in fact changed. Without the Temple, there was no way of dealing with sin in the "old way"; this proved what early Jesus Followers claimed, that Jesus was the New Covenant. Jesus was the canary in the coal mine for the Second Temple's destruction.
Again, this last part is important because without understanding what the Temple represents and how that relates to Jesus's death and the Temple's destruction, you will never understand why he had to die. He had to die to become the Temple that could not be destroyed. The physical building was gone because the final Sacrifice had already happened, rendering the old system obsolete.
So, in simplest terms: Jesus died for our sins because the world before Jesus operated on a blood debt from Cain and Abel that the Priesthood washed away in the Temple ritualistically with a physical lamb being sacrificed. The Jews, being God's chosen people, were the only ones who understood this method of washing away their sins. Jesus takes the place of the Lamb and extends the atonement not just to the Jews, but to all the children of Noah, breaking the curse of Cain, and freeing humanity from their ancient blood curse.