Why was her death a fixed point?
It's not entirely clear. It would seem that with unstable points, you can change something and time will heal, whether to a somewhat different reality (for want of a nail) or to the same reality (in spite of a nail).
When the Doctor saved Adelaide from the Martian nuclear water zombies (a phrase I never thought to hear myself say), she committed suicide, and set history back on exactly the path it had been on. Time healed itself.
When the Dalek Emperor started taking control of Earth through the Jagrafess, the Doctor encountered a timeline where Earth's development had been held back significantly. Time healed itself, but to a different reality.
Some points, however, cannot be changed. What this apparently means is that if one messes with them, all of history will start happening at once, and then the universe will unravel, as seen in The Wedding of River Song.
In that context, consider the following script:
ASHILDR: Still no pulse?
CLARA: Time isn't healing. I am still
ASHILDR: You know what that means?
CLARA: It means my death
is a fixed event. The universe depends on it happening.
It is unclear what makes something a fixed point, so the first part of the question cannot be answered. But as to why it couldn't be changed, if it were to be changed time would fail to heal itself around her. If time had healed around Clara, she would have come back to life, properly, and another course of events would have proceeded. Since it did not (as indicated by her continued lack of a pulse), it indicated that her death must be a fixed point. Swapping out a double simply would not "work," because her death is what is fixed, not someone's perception of it.
We can't really answer the why, only the "how do we know" and "why can't it be changed."
Note that Clara cheated. She has to die in that alley at some point in her personal timeline, but as to when...who knows? So she can have as much time as she wants. It's a very Doctor-ish loophole, isn't it?